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# Theoretical and Methodological Developments for Markov Chain Monte Carlo Algorithms for Bayesian Regression

## Material Information

Title: Theoretical and Methodological Developments for Markov Chain Monte Carlo Algorithms for Bayesian Regression
Physical Description: 1 online resource (94 p.)
Language: english
Creator: Roy, Vivekananda
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

## Subjects

Subjects / Keywords: bayesian, da, efficiency, markov, monte, multivariate, probit, px, regenerative
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: I develop theoretical and methodological results for Markov chain Monte Carlo (MCMC) algorithms for two different Bayesian regression models. First, I consider a probit regression problem in which $Y_1,\dots,Y_n$ are independent Bernoulli random variables such that $\Pr(Y_i =1) = \Phi(x_i^T \beta)$ where $x_i$ is a $p$-dimensional vector of known covariates associated with $Y_i$, $\beta$ is a $p$-dimensional vector of unknown regression coefficients and $\Phi(\cdot)$ denotes the standard normal distribution function. I study two frequently used MCMC algorithms for exploring the intractable posterior density that results when the probit regression likelihood is combined with a flat prior on $\beta$. These algorithms are Albert and Chib's data augmentation algorithm and Liu and Wu's PX-DA algorithm. I prove that both of these algorithms converge at a geometric rate, which ensures the existence of central limit theorems (CLTs) for ergodic averages under a second moment condition. While these two algorithms are essentially equivalent in terms of computational complexity, I show that the PX-DA algorithm is theoretically more efficient in the sense that the asymptotic variance in the CLT under the PX-DA algorithm is no larger than that under Albert and Chib's algorithm. A simple, consistent estimator of the asymptotic variance in the CLT is constructed using regeneration. As an illustration, I apply my results to van Dyk and Meng's lupus data. In this particular example, the estimated asymptotic relative efficiency of the PX-DA algorithm with respect to Albert and Chib's algorithm is about 65, which demonstrates that huge gains in efficiency are possible by using PX-DA. Second, I consider multivariate regression models where the distribution of the errors is a scale mixture of normals. Let $\pi$ denote the posterior density that results when the likelihood of $n$ observations from the corresponding regression model is combined with the standard non-informative prior. I provide necessary and sufficient condition for the propriety of the posterior distribution, $\pi$. I develop two MCMC algorithms that can be used to explore the intractable density $\pi$. These algorithms are the data augmentation algorithm and the Haar PX-DA algorithm. I compare the two algorithms in terms of efficiency ordering. I establish drift and minorization conditions to study the convergence rates of these algorithms.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Vivekananda Roy.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022377:00001

## Material Information

Title: Theoretical and Methodological Developments for Markov Chain Monte Carlo Algorithms for Bayesian Regression
Physical Description: 1 online resource (94 p.)
Language: english
Creator: Roy, Vivekananda
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

## Subjects

Subjects / Keywords: bayesian, da, efficiency, markov, monte, multivariate, probit, px, regenerative
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: I develop theoretical and methodological results for Markov chain Monte Carlo (MCMC) algorithms for two different Bayesian regression models. First, I consider a probit regression problem in which $Y_1,\dots,Y_n$ are independent Bernoulli random variables such that $\Pr(Y_i =1) = \Phi(x_i^T \beta)$ where $x_i$ is a $p$-dimensional vector of known covariates associated with $Y_i$, $\beta$ is a $p$-dimensional vector of unknown regression coefficients and $\Phi(\cdot)$ denotes the standard normal distribution function. I study two frequently used MCMC algorithms for exploring the intractable posterior density that results when the probit regression likelihood is combined with a flat prior on $\beta$. These algorithms are Albert and Chib's data augmentation algorithm and Liu and Wu's PX-DA algorithm. I prove that both of these algorithms converge at a geometric rate, which ensures the existence of central limit theorems (CLTs) for ergodic averages under a second moment condition. While these two algorithms are essentially equivalent in terms of computational complexity, I show that the PX-DA algorithm is theoretically more efficient in the sense that the asymptotic variance in the CLT under the PX-DA algorithm is no larger than that under Albert and Chib's algorithm. A simple, consistent estimator of the asymptotic variance in the CLT is constructed using regeneration. As an illustration, I apply my results to van Dyk and Meng's lupus data. In this particular example, the estimated asymptotic relative efficiency of the PX-DA algorithm with respect to Albert and Chib's algorithm is about 65, which demonstrates that huge gains in efficiency are possible by using PX-DA. Second, I consider multivariate regression models where the distribution of the errors is a scale mixture of normals. Let $\pi$ denote the posterior density that results when the likelihood of $n$ observations from the corresponding regression model is combined with the standard non-informative prior. I provide necessary and sufficient condition for the propriety of the posterior distribution, $\pi$. I develop two MCMC algorithms that can be used to explore the intractable density $\pi$. These algorithms are the data augmentation algorithm and the Haar PX-DA algorithm. I compare the two algorithms in terms of efficiency ordering. I establish drift and minorization conditions to study the convergence rates of these algorithms.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Vivekananda Roy.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022377:00001

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6afdb395dd0612c6d3e2d8013928be758969ba68

THEORETICAL AND METHODOLOGICAL DEVELOPMENTS FOR MARK(OV
CHAIN MONTE CARLO ALGORITHMS FOR BAYESIAN REGRESSION

By
VIVEK(ANANDA ROY

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

S2008 Vivekananda Roy

To my parents

ACKNOWLEDGMENTS

I extend my sincerest thanks to my advisor .Jint Hohert for his guidance throughout

my graduate study at University of Florida. I feel fortunate to have .Jint as my PhD

advisor. His guidance, help, enthusiasm, were all crucial to making this thesis take its

current shape. I'ni deeply grateful to hint for many other things, not the least for his

inspiring words in my hours of need.

I would also like to thank Professors Ben Bolker, Hani Doss and Brett Presnell for

agreeing to serve on my coninittee. I am particularly grateful to Professors Hani Doss and

Brett Presnell for being so kind to me over the past five years. I learned not only statistics

but also a lot about Emacs, LaTex and R front them.

I would also like to thank Professors Bob Dorazio, Malay Ghosh and Andrew

Rosalsky for sparing a lot of their valuable time on academic discussions with me and

giving me advice on several issues. I thank all my teachers from school, college and Indian

Statistical Institute whose dedication to teaching and quest for knowledge have inspired

me to pursue higher study.

Special thanks go to All .Ilv:adi and Parag for their friendship, care and support. I

have learned a lot about life in past five years front both of them. I owe deep gratitude to

.Jethinia whose care and affection I will never forget. I am thankful to Shuva whose love

and enthusiasm for niathentatics have ahr-l- .- inspired me.

I am indebted to many other people, mostly front my village, who guided me and

encouraged me during the formative years of my life: Arunda, Arunkaku, Bapida,

Bonikeshjethu, Budhujethu, Shashankajethu and my uncle.

Finally, I would like to thank my parents for ahr-l-~ .14eing a driving force in my life.

I often feel that whatever I have achieved is only due to nly parents' sacrifice, hard work

and honesty.

page

ACK(NOWLEDGMENTS .......... . .. .. 4

LIST OF TABLES ......... ... . 6

ABSTRACT ............ .......... .. 7

CHAPTER

1 INTRODUCTION ......... ... .. 9

2 MARKOGV CHAIN BACKGROUND . ..... .. 18

3 BAYESIAN PROBIT REGRESSION . ..... .. 24

3.1 Introduction .. .. ... . .. ..... .. 24
3.2 Geometric Convergence and CLTs for the AC Algorithm .. .. .. 27
3.3 Comparing the AC and PX-DA Algorithms .... .. 35
3.4 Consistent Estimators of Asymptotic Variances via Regeneration .. .. 37

4 BAYESIAN MULTIVARIATE REGRESSION .. .. .. .. 48

4.1 Introduction ......... . .. .. 48
4.2 Proof of Posterior Propriety ........ ... .. 50
4.3 The Algorithms ......... . .. 58
4.3.1 Data Augmentation ....... .. .. 58
4.3.2 Haar PX-DA Algorithm . ..... .. .. 60
4.4 Geometric Ergodicity of the Algorithms ... .. .. .. 61

5 SPECTRAL THEOREM AND ORDERING OF MARKOGV CHAINS .. .. 71

5.1 Spectral Theory for Normal Operators .... ... 71
5.2 Application of Spectral Theory to Markov C'!s I!us< ... .. .. 81

APPENDIX: CHEN AND SHAO'S CONDITIONS ... .. .. 89

REFERENCES ......._._.. ........_._.. 90

BIOGRAPHICAL SK(ETCH ....._._. .. .. 94

LIST OF TABLES

Table

page

3-1 Results based on R = 100 regfenerations ...... .. . 47

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THEORETICAL AND METHODOLOGICAL DEVELOPMENTS FOR MARK(OV
CHAIN MONTE CARLO ALGORITHMS FOR BAYESIAN REGRESSION

By

Vivekananda Roy

August 2008

C'I I!r: James P. Hohert
Major: Statistics

I develop theoretical and methodological results for Markov chain Monte Carlo

(jl\C'l\C) algorithms for two different B li-. -1 Ia regression models. First, I consider a

profit regression problem in which }*\,...,Y*, are independent Bernoulli random variables

such that Pr(}* = 1) = #(:rf#) where :rs is a p-dimensional vector of known covariates

associated with }*, /3 is a p-dimensional vector of unknown regression coefficients and

#(-) denotes the standard normal distribution function. I study two frequently used

MC1| C algorithms for exploring the intractable posterior density that results when the

profit regression likelihood is combined with a flat prior on 73. These algorithms are

Albert and Chih's data augmentation algorithm and Liu and Wu's PX-DA algorithm.

I prove that both of these algorithms converge at a geometric rate, which ensures the

existence of central limit theorems (CLTs) for ergodic averages under a second moment

condition. While these two algorithms are essentially equivalent in terms of computational

complexity, I show that the PX-DA algorithm is theoretically more efficient in the sense

that the .I-i-ini!!lle'~ variance in the CLT under the PX-DA algorithm is no larger than

that under Albert and Chih's algorithm. A simple, consistent estimator of the .I-i-inia..'l~e

variance in the CLT is constructed using regeneration. As an illustration, I apply my

results to van Dyk and Meng's lupus data. In this particular example, the estimated

.I-i-inidul'lic relative efficiency of the PX-DA algorithm with respect to Albert and Chih's

algorithm is about 65, which demonstrates that huge gains in efficiency are possible by

using PX-DA.

Second, I consider multivariate regression models where the distribution of the errors

is a scale mixture of normals. Let xr denote the posterior density that results when the

likelihood of n observations from the corresponding regression model is combined with

the standard non-informative prior. I provide necessary and sufficient condition for the

propriety of the posterior distribution, xr. I develop two 1\C1| C algorithms that can he

used to explore the intractable density xr. These algorithms are the data augmentation

algorithm and the Haar PX-DA algorithm. I compare the two algorithms in terms of

efficiency ordering. I establish drift and minorization conditions to study the convergence

rates of these algorithms.

CHAPTER 1
INTRODUCTION

Realistic statistical modeling often leads to a complex, high-dimensional model that

precludes analytical, closed-form calculation which is required for statistical inference and

prediction. If we combine the complex model with a prior distribution on the unknown

parameters, as is done in B li-o -1 .Is Statistical analysis, the result is typically an intractable

posterior distribution of the model parameters given the observations. Suppose xr(0|y)

is the posterior density of the p x 1 vector of unknown model parameters, 8, given

the observations, y. In B li-, -i Ia inference, we are often interested in evaluating the

expectation of some function, my- f, with respect to the posterior density xr, i.e., we want

to know

Ex f= f 0)x(|y) O .(1-1)

Because the density xr(0|y) is a complicated function, closed-form calculation of the

above integral is generally impossible. We assume that the above integral exists and is

finite. Since Ex f can not be evaluated analytically, we use either deterministic numerical

integration techniques or simulation based methods to get an approximate value of (1-1).

Before delving into these computational methods, we provide two motivating examples.

In both of these examples, statistical modelling results in an intractable posterior density

making explicit closed-form calculation of the corresponding posterior expectations

impossible.

In problems involving toxicity tests and '.1. .-- li experiments, the responses are

often binary since what is observed is whether the subject is dead or whether a tumor

has appeared. A popular method of analyzing binary data is through B li-o -1 .Is analysis

with a probit link function. Suppose that we observe n independent Bernoulli random
variables, Yi,...,Y,, and we assume that Pr( = 1) = #(xf ) where x4 is ap xn 1 vectonr of

known covariates associated with ~, is a p x 1 vector of unknown regression coefficients

and #(-) denotes the standard normal distribution function. For ye {0(, 1}", that is,

Y = (Y1,... ,ys~) and y, E {0,1}), we have

i= 1

If we use a flat prior on p, the marginal density of the data takes the form

It is not obvious whether cy (y) < 00. We address this issue in chapter 3. Assuming

city) < 00, the posterior density of p takes the following form

i= 1

Clearly, the posterior density xr(P | y) is too complicated to allow explicit closed-form

calculation of posterior expectations of functions of P.

It has long been known that heavy-tailed error distributions are often required

when modelling financial data. Specific scale mixtures of normal distributions can be

used for modelling heavy-tailed data. Our second example is a B li- Ion multivariate

regression model where the distribution of the errors is a scale mixture of normals.

Suppose Y1, Y2, *, n are d-dimensional random vectors (e.g. returns on some assets)

satisfying the linear regression model

S= PXi + Ei

where p is the k x d matrix of unknown regression coefficients, xi's are k x 1 vectors of

known explanatory variables and we assume that, conditional on the positive definite

matrix E, the d-variate error vectorS E1, ,E are independently and identically

distributed with common density

fH6d E)= exp E ETZ-1E dUH(5
o(2xr) |ECa

where H(-) is the distribution function of a non-negative random variable. The density,

fH, clearly, iS a multivariate scale mixture of normals. The density fH can be made

heavy-tailed by choosing H appropriately.

We can rewrite the above regression model as

Y = Xp +E

where Y = (Yi, .. ., Y,)T is the a x d matrix of observations, X = (xl, xa2, *, )T iS the

a x k matrix of covariates and E = (E, .. En) iS the a x d matrix of error variables. The

likelihood function for this regression model is given by

f y|, ) (x)|E exp 6 -0x 2 H6

If we consider the standard noninformative prior on (P, E), i.e., if we assume that the prior
d~l
density, xr(P, E) oc |E| 2 the posterior density takes the following form

C2 1 x "

C2(y i i=1 0~ (2xr) |E|-I 2

where c2 9) is the marginal density of y given by

d(d+1)
where W CR 2W is the set of d x d positive definite matrices. In chapter 4, we

provide necessary and sufficient conditions for c2 9) < OO. As in the previous example,

posterior expectations with respect to the posterior density, xr(P, E | y), are not available in

closed-form.

We now discuss different computational methods that can be used to approximate

(1-1). These computational methods are broadly of two types, namely, numerical

integration methods and simulation based methods. If the dimension, p, is not large,

numerical integration techniques can be efficiently used to obtain a good approximation of

(1-1). But, as p increases, numerical integration techniques become less and less efficient

because of the well known problem called the curse of I.:Is:.: ,:r.......lU;, In this dissertation,

we consider simulation based methods to estimate the posterior expectations.

An alternative to numerical integration is to estimate (1-1) by Monte Carlo sampling.

Monte Carlo integration requires drawing iid samples X*,X*,...,XA_, from xr(-) and

then using the sample mean
m-1
fm f(X J ),
j=0
to estimate the expectation (population mean) in (1-1). The justification of Monte Carlo

methods comes from the strong law of large numbers (SLLN~), which guanrantees that fm

converges almost surely to Ex f as m tends to infinity. So, Ex f can be well approximated

by fm provided the sample size, m71, is large enough. We often know r only up to its

normalizing constant, i.e., usually, ,, xr(0|y) de is unknown. In that case, we can use

rejection sampling methods Robert and Casella [38, C'!s Ilter 2.3] to obtain an iid sample

from xr. Rather than giving details about different Monte Carlo methods, we now address

an important issue that the experimenters ahr-l- .- face -i.e., how to choose the agi

sample size, m?

How large a sample size is sufficient is a subjective matter. It depends on how much

error we are willing to accept in the approximation. One way to measure the accuracy

in the approximation is by the width of a 95' confidence interval for Ex f. A confidence

interval for Ex f can be obtained using the central limit theorem (CLT) for the estimator

fm. If f has a finite second moment with respect to r, i.e., if Exf2 < 00, then by the
classical central limit theorem we have

fl7, Ex f 1 N(0,v2) aS n 4 OO

where v2 = Exf2 (Exrf)2. An .l-i-mptotic 95' confidence interval for Ex f is given by

Im + 2sm,/~ where s,, is a~ strongly con~sistent, estim~a~tor of v2" giVen? by
m-1

i=0

Since the sample size, m, is under our control, the main benefit of calculating the

confidence interval is to determine whether the Monte Carlo sample size we choose is

large enough. In practice, one draws a random sample of size M~ from xr for some finite

n~umber M! anld constructs th~e confidence interval fMI + 28M/2il. If the Ilength? of the

resulting confidence interval, 4sM/ lz, Seems to be satisfactory, then one stops the

sampling and reports fM RS an eStimate of Ex f. On the contrary, if the .I-i-mptotic 95' .

confidence interval is deemed too wide, then M~ can be increased appropriately and further

simulation can be carried out until the desired level of accuracy is achieved. Of course,

in the latter case, if we know beforehand the precision, e, that we want to achieve then

we can use SM/ RS a pilot estimate of v to calculate an approximate sample size, namely,

(28M @l~2, that we need.

In practice, making iid draws from xr might not be feasible. For example, in the probit

regression model that we mentioned, it is difficult to make iid draws from the posterior

density, xr(Ply), especially when the dimension, p, is large. Similarly, for the multivariate

regression model that we discussed, it is problematic to produce a useful Monte Carlo

method to simulate from the posterior density, xr(P, Ely).

Surprisingly, it is straightforward to construct a Markov chain with stationary

distribution xr even when direct simulation from xr is impossible. As explained in the next

paragraph, it turns out that it is indeed possible to approximate (1-1) by simulating a

Markov chain with stationary distribution xr. This is the basic principle of Markov chain

Monte Carlo (ifi'lC) method. The most general algorithm for producing Markov chains

with arbitrary stationary distribution xr is the Metropolis-Hastings (il-H) algorithm. A

simple introduction to the M-H algorithm is given in Chib and Greenberg [7]. Another

widely used MCijlC algorithm is the Gibbs Sampler [4]. Suppose the p-dimensional vector

8 in (1-1) can be written as 8 = (01, 82, ). The simplest Gibbs sampler (but, not

the general Gibbs sampler) requires one to be able to simulate from all univariate full

conditional densities of xr i.e., it is required to simulate from the conditional distributions,

s| {0;, j / i} for i = 1, 2,..., p. It is also possible to create a hybrid algorithm which uses

different versions of M-H algorithm together with Gibbs sampler to construct a Markov

chain with stationary distribution xr. As our discussion so-----~ -r- there is a plethora

of Markov chains with stationary distribution xr. In order to choose between MCijlC

algorithms, we need an ordering of Markov chains having the same stationary distribution

xr. In C'!s Ilters 2 and 5, we describe different partial orderings of Markov chains.

Let {Xjy}Ro denote the Markov chain associated with an MCijLC algorithm that is

used to explore xr. If {Xjy}R is Harris ergodic (defined in C'!s Ilter 2), the ergodic theorem

implies that, no matter what the distribution of the starting value, Xo,
m-1
fm := ~f(X,)
j=0

is a strongly consistent estimator of Ex f, i.e., fm Ex f almost surely as m c o. So,

the ergodic theorem (like the SLLN in iid case) ensures that Ex f can be approximated

by running a well behaved Markov chain for sufficiently large number of iterations. In

practice, as in iid case, one simulates the chain for a finite number of iterations, ;?i M~',

and reports fM;r aS the estimate of Ex f. Suppose there is an associated central limit

theorem (CLT) given by

fm x f (0, .2) S 0 00 (1-)

and that we have a consistent estimator of o.2, Call it ~2. So, we can compute an

.I-i-inidlli'lc standard error for fM;r, Which is given by &/ ~l. As in the iid case, the

.I-i-inidlli'lc 95'. confidence interval given by fMr & 29-/21 can then be used to decide

whether there is any need for further simulation.

As -II_0-r-- -1. in the previous paragraph, establishing a central limit theorem for a

Markov chain is essential in order to put MC10L on equal footing with iid sampling.

Unfortunately, unlike in classical Monte Carlo methods, the finite second moment

condition i.e., E, f2 < 00 does not insure a CLT for f,~. In addition, the Harris ergodicity

which establishes the strong consistency of foz is not enough to guarantee that (1-2)

holds. It generally requires rigorous analysis of the Markov chain {Xj;}Ro in order to

prove that CLT holds for f,,. There are several v- .--s Of establishing the CLT in (1-2).

These approaches can he broadly divided into two categories. One approach is based on

probabilistic (convergence rate) aI, l1i--;-; of the Markov chain. We give a brief description

of these techniques in C'!s Ilter 2. The other approach exploits results from functional

analysis (see Chapter 5).

Another difficulty in constructing the confidence interval, fn/r & 2o-/ W1 is that even

when there is a CLT, finding a simple, consistent estimator of the .I-i-mptotic variance,

0.2, can he challenging due to the dependence among the random variables in the Markov

chain. Mykland, Tierney, and Yu [:33] show that when CLT exists, regenerative simulation

(R S) methods can he used to construct a consistent estimator of o.2 by uncovering the

regenerative properties of the Markov chain (Section :3.4). The regenerative simulation

technique basically breaks the whole Markov chain up into iid pieces (tours) by keeping

track of the regeneration times. Then, standard iid theory can he used to analyze the

.I-i-ini!!lle'~ behavior of the ergodic average, f,,2 and thus a simple, consistent estimator

of .l-i-!Injdull- variance is obtained. It might not he easy to implement the RS method in

practice. There are other methods like batch mean and spectral methods which are easier

to employ to estimate the .I-i-inidull-; 1 variance (Jones et al. [21] and the references cited

therein). The advantage of using RS method is that it is on stronger theoretical footing

than the other methods.

We now provide a brief overview of the four remaining chapters of this dissertation.

In the next chapter, we review some results from general state space Markov chain

theory. In particular, we mention sufficient conditions for Markov chain CLT and provide

a partial ordering of Markov chains based on their performance in the central limit

theorem.

In (I Ilpter 3, we study two MC \!C algorithms that are frequently used for exploring

the posterior density, xr(Ply), that we mentioned before in the context of the probit

regression example. These algorithms are Albert and Chib's [1993] data augmentation

algorithm and Liu and Wu's [1999] PX-DA algorithm. We study the convergence rate

of these algorithms and prove the existence of central limit theorems (CLTs) for ergodic

averages under a second moment condition. We compare these two algorithms and show

that the PX-DA algorithm should ah-- .--s be used since it is more efficient than the

other algorithm in the sense of having smaller .I-i-mptotic variance in the central limit

theorem (CLT). A simple, consistent estimator of the .I-i-inidllicl~ variance in the CLT is

constructed using regenerative simulation methods.

In C'!s Ilter 4, we consider B li-, -i Ia multivariate regression models where the

distribution of the errors is a scale mixture of normals. We noticed before that if

the standard noninformative prior is used on the parameters (P, E), then posterior

expectations with respect to the corresponding posterior density, xr(P, Ely), are not

available in closed-form. We develop two MC1| C algorithms that can be used to explore

the density xr(P, Ely). These algorithms are the data augmentation algorithm and the

Haar PX-DA algorithm. We compare the two algorithms and study their converge rates.

We also provide necessary and sufficient conditions for the propriety of the posterior

density, xr(P, Ely).

While in ChI Ilpters 3 and 4, we used probabilistic techniques to 2.1, lli-. .. different

MC1| C algorithms, it is possible to take a functional analytic approach to study and

compare different Markov chains. In ChI Ilpter 5, we give a brief overview of some results

from functional 2.!! l1i--;- In particular, we discuss the spectral theorem for bounded,

normal operators on Hilbert space. We show how these results of functional analysis can

be used to study Markov chains.

CHAPTER 2
MARK(OV CHAIN BACKGROUND

Let A = {Ami~o denote a time-homogeneous discrete-time Markov chain on a

general state space X equipped with a countably generated o--algebra B(X). Let Pm(x, A)
be the m-step Markov transition function associated with A for m = 1, 2, 3,. .. So,

Pm(x, A) denotes the probability that the Markov chain at x will be in the set A after m

steps (transitions), that is, for x E X, Ae B (X) and le {0(, 1, 2, ...},

Pm(x, A) = Pr(Am+i E A | At = x)

When m = 1, we simply denote the one step Markov transition function by P(x, A) and

for m = 2, 3, .. ., Pm(x, A) is defined iteratively by

Pm~x A) P~, dyPm-(y, A).

A probability measure xr on B(X) is called an invariant probability measure for A, if, for all

measurable sets A,

xr(A) = P,, AxIdx)

Note that,

CP2(x, A)a(dx:) = SP(x.)P, d)P(y, )=~x = (dy)P(y,l A)= (A).

Similarly, we can show that f Pm(x, A) x (dx) = x (A) for m = 1, 2, 3,. .. So, if Ao ~ xr,

then Am ~ xr for all m and A is stath-.: o a,~r in distribution.

Let L2(;,) be the vetor space of real-valued, measurable functions on X that are

square-integrable with respect to xr, i.e.,

The Markov chain A is said to be reversible with respect to xr if for all functions f, g e

L2(r

fC S (yl~gr)P(x, dy)r(dx) = f~S I(x)g(y)P(x, dy)i(dxl).

If we take g(x) = 1 in the above equation, we get

f S~(y)P(x, dy~~h)=XS(dx) =f(x)Ix dy)i(dx) = f l~(x:)xd)

i.e., xr is invariant for A. So, if a Markov chain is reversible with respect to xr then xr is
invariant for the chain.

Suppose that p is a non-trivial, o--finite measure on X. The Markov chain A is
called p-irreducible if for each x E X and each set A with p(A) > 0, there exists an

m EN := {1, 2,. .. }, which may depend on x and A, such that Pm(x, A) > 0. In words,

the chain is p-irreducible if every set with positive p measure is accessible from every

point in the state space. The measure p is called an irr..I;. 09ii;, measure for A. As in
M.~ in and Tweedie [30, Section 4.2], when we ;?i "A is ~-irreducible" we mean that A\

is p--irreducible for some p and that is a maximal irre 7 09ii;, measure for A. Two

properties of maximal irreducibility measures that will be used in the sequel are (i) if p is
an irreducibility measure and is a maximal irreducibility measure, then p is absolutely

continuous with respect to (denoted > p), and (ii) a maximal irreducibility measure is

unique up to equivalence, i.e., if I1 and 2a are both maximal irreducibility measures, then

#1 > #2 and 2a > 1 (denoted 21 a

The p--irreducible Markov chain A is ap~eriodic if there do not exist an integer d > 2

and disjoint subsets Ao, Al,. ., Ad- _c X with p~(Ao) > 0, such that for all i = 0, 1, .. d- 1

and all x E Ai,

P(x, Aj) = 1 for j = i + 1(mod d).

Suppose A is ~-irreducible and define B+(X) = {A E B(X) : ~(A) > 0}. The Markov

chain A is called Harris recurrent if for all Ae B +(X),

Pr (Am EA i.o. | Ao = x) = 1 for all x E X.

The Markov chain A is called Harris ergodic if it is ~-irreducible, periodic and Harris

recurrent. The Harris ergodicity of a Markov chain is often easy to verify in practice and it

implies that, for every x E X,

||Pm(x, -) jT(-) | 0a m co ., ,

where ||Pm(x, -) xr(-)|| denotes the total variation distance between the probability

measures Pm"(x, -) and 7i(-), i.e., the supremumn over measurable A of Pm"(x, A) x(A) .

However, the Harris ergodicity tells us nothing about the rate at which this convergence

takes place. If it takes place at a geometric rate, then A is said to be geome/0.:. ellol ergodic.

More precisely, the Harris ergodic Markov chain A is geometrically ergodic if there exists a

constant p E [0, 1) and a function M : X [0, 00) such that for any x E X and any me N ,

|| P~x,-) -x () | < Mx) m .(2-1)

We now describe methods that are used to prove the geometric ergodicity of a Markov

chain. One method of proving that A is geometrically ergodic is by establishing drift

and minorization conditions. There are several v-wsi~ of doing this (11. i-n and Tweedie

[31], Rosenthal [45], Roberts and Tweedie [40]). Here, we describe a method based on

Rosenthal's [1995] work.

A drift condition holds if for some function V : X [0, 00),

PV < AV + L

for some A E [0, 1) and some L < 00, where (PV)(x) = f, V(y)P(x, dy). The function V is
often called a drift function.

An associated minorization condition holds if for some probability measure Q(-) on

B(X) and some E > 0 we have

P(x, A) > EQ(A) VX E C and VA E B(X)

where C := {x E X : V(x) < l} with I being any number larger than 2L/(1 A).

Rosenthal's [1995] Theorem 12 shows that the above drift and minorization conditions,

together, imply that A is geometrically ergodic. In Chapter 4, we employ drift and

minorization conditions to prove the geometric ergodicity of the data augmentation

algorithm used in B li-, Io multivariate Student's t regression problem.

One advantage of proving geometric ergodicity of A by establishing the above drift

and minorization conditions is that using Rosenthal's [1995] Theorem 12, we also can

calculate an upper bound of M~(x)pm in (2-1). This upper bound can be used to compute

an appropriate burn-in period ( Jones and Hobert [23], Marchev and Hobert [28]). There

are other methods of proving geometric ergodicity of a Markov chain that do not provide

any quantitative bound of M~(x)pm in (2-1). We describe one such method now.

We will assume that X is equipped with a locally compact, separable, metrizable

topology with B(X) as the Borel o--field. A function V : X [0, 00) is said to be

unbounded of compact sets if for every y > 0, the level set {x : V(x) < y} is compact.

The Markov chain A is said to be a Feller chain if, for any open set Oe B (X), P(-, O) is

a lower-semicontinuous function. The following proposition is a special case of M.~ i-n and

Tweedle's [1993] Lemma 15.2.8.

Proposition 1. Sup~pose that the Harris ergodic M~arkov chain A is a Feller chain.

Supplose further that the support of a maximal irrech;.-.l..7. ;, measure has non-tii pl ;

interior. If for some V :X [0, 00) that is unbounded of compact sets

PV < AV + L

for some A E [0, 1) and some L < 00, then the M~arkov chain, A, is geomen,... ellol ergodic.

In C'!s Ilter 3, we apply Proposition 1 to establish geometric ergodicity of MC \!C

algorithms used in B li-, -i Ia probit regression problem. Hobert and G. o;r [15] emploi-. I

Proposition 1 to establish the geometric ergodicity of Gibbs samplers associated with
B li-, -i Ia hierarchical random effects models.

Notice that, unlike Proposition 1, the drift condition in Rosenthal's [1995] Theorem

12 does not require the drift function, V, to be unbounded off compact sets. Also,

Rosenthal's [1995] Theorem 12 does not need A to be a Feller chain.

The driving force behind MCijlC is the ergodic theorem, which is simply a version

of the strong law that holds for well-behaved Markov chains, e.g., Harris ergodic Markov

chains. Indeed, suppose that f : X R I is such that |x Ifldx < 00 and define Exf =

f dx The theergoic teore .v that the average fm = m-] CE f(As) converges

almost surely to Ex f no matter what the distribution of Ao. This justifies our use of fm as

an estimator of Ex f. We will ;?i that there is a CLT for fm if there exists a a2 E (0, 00)

such that, as m 00o,

L(f m Ex f) iN~(0, a"2

As explained in OsI Ilpter 1, CLTs are the basis for .I-i-inidllicl~ standard errors, which can

be used to ascertain how large a sample is required to estimate Ex f. Unfortunately, while

the Harris ergodicity of a Markov chain does imply that the ergodic theorem holds, this

is not enough to guarantee the existence of CLTs. However, if A is geometrically ergodic

and reversible with respect to xr, then the CLT holds for every f such that S, f2d~ o

that is, for every f E L2(;T) [41]. (For more on the CLT in MCil C, see C'I I>. and Geyer

[5], Mira and Geyer [32], Jones [20] and Jones et al. [21].) For a thorough development of

general state space Markov chain theory, see Nummelin [34] and M.~ i-n and Tweedie [30].
Robert and" Rosentha^l [43] provides a concise, self-contained description on general state

space Markov chains (also see Tierney [49]).

As mentioned in OsI Ilpter 1, for a given distribution function, xr, there are large

number of MCil C algorithms with stationary distribution xr. One way to order these

algorithms is based on their performance in CLT. Note that the .I-i-md.l!lle~; variance, O.2

in (1-2) depends both on the function f and the particular MC1| C algorithm that we

are using. Suppose P and Q he the Markov transition functions corresponding to two

different MCijlC algorithms with stationary distribution xr. Let us denote O.2 for these

two algorithms by n( f, P) and v( f, Q) respectively. Assume, both v( f, P) and v( f, Q) are

finite. Then if we are interested in calculating E, f, we prefer the Markov chain P over Q

if v(f, P) < v(f, Q) provided the two chains are equivalent in terms of simulation effort.

On the other hand, if we do not assume any prior knowledge about the function whose

expectation we want to evaluate, we need a uniform ordering as below.

Definition 1. [SE] If P and Q
Markov chains with intericent Igo~~~l.:.:sl..;i measure x., then P is better than Q in the

e~ff~.,.. I ordering written P FE Q. if c(f, P) < 'U(f, Q) for every f E L2(,)

In ChI Ilpter 3 and ChI Ilpter 4, we order different MCijlC algorithms in terms of

efficiency ordering.

CHAPTER 3
BAYESIAN PROBIT REGRESSION

3.1 Introduction

Suppose that Yi,...,Y, are independent Bernoulli random variables such that

Pr(~ = 1) = #(xTP) where xi is a p x 1 vector of known covariates associated with ~, P

is a p x 1 vector of unknown regression coefficients and #(-) denotes the standard normal

distribution function. For ye {(, 1}", that is, y = (yl,..., y,) and yi E {0,1}), we have

i= 1

A popular method of making inferences about P is through a B li-o -1 .I analysis with a flat

prior on p. Define the marginal density of the data as

C'I. in and Shao [6] provide necessary and sufficient conditions on y and {xi}", for

city) < 00 and these conditions are stated explicitly in the Appendix. When these

conditions hold, the posterior density of P is well defined (i.e., proper) and is given by

Unfortunately, the posterior density xr(P | y) is intractable in the sense that expectations

with respect to it, which are required for B li-, -i Ia inference, cannot be computed in closed

form. Moreover, as we mentioned in (I Ilpter 1, classical Monte Carlo methods based

on independent and identically distributed (iid) samples are difficult to apply when the

dimension, p, is large. These difficulties spurred the development of Markov chain Monte

Carlo methods for exploring xr(Ply). The first of these was Albert and Chib's [1993] data

augmentation algorithm, which we now describe.
Let X denote the, \x p design matrix whose ith row is x { and, for z = (zi, ..., z,)T E

RW", let p = P(z) = (XTX)-1XTz. Also, let TN(p, is2, w) denote a normal distribution with

mean p and variance is2 that is truncated to be positive if w = 1 and negative if w = 0.

Albert and Chib's algorithm (henceforth, the "AC als..i s~I lIn~ ) simulates a Markov chain

whose invariant density is xr(P | y). A single iteration uses the current state P to produce

the new state p' through the following two steps:

(i) Draw zi, .., z, independently with ze ~ TN(x'P, 1, yi)

(ii) Draw p' ~ N,( i(z), (XTX)-1)
Albert and Chib [1] has been referenced over 350 times, which shows that the AC

algorithm and its variants have been widely applied and studied.

The PX-DA algorithm of Liu and Wu [27] is a modified version of the AC algorithm

that also simulates a Markov chain whose invariant density is xr(P | y). A single iteration of

the PX-DA algorithm entails the following three steps:

(i) Draw zi, .., z, independently with ze ~ TN(x'P, 1, yi)

(ii)~ ~ ~ ~ 1 Dra g2l ~ am liz x(XI1XI)I; 1Xz)2 and set z' =(z,.,g,
(iii) Draw ii' ~ Np( j(zj), (X X)-1)
Note that the first and third steps of the PX-DA algorithm are the same as the two steps

of the AC algorithm so, no matter what the dimension of P, the difference between the

AC and PX-DA algorithms is just a single draw from the univariate gamma distribution.

For typical values of n and p, the effort required to make this extra univariate draw

is insignificant relative to the total amount of computation needed to perform one

iteration of the AC algorithm. Thus, the two algorithms are basically equivalent from

a computational standpoint. However, Liu and Wu [27] and van Dyk and Meng [51] both

provide considerable empirical evidence that autocorrelations die down much faster under

PX-DA than under AC, which so-----~ -r- that the PX-DA algorithm "mixes f I-I. I than the

AC algorithm. (Liu and Wu [27] also established a theoretical result along these lines see

the proof of our Corollary 1.)

Suppose we require the posterior expectation of f (P) given y, i.e., we want to know

assuming this integral exists and is finite. Let {@}Rj"o denote the Markov chain associated

with either the AC or PX-DA algorithm. We later show in this chapter that {@}Rj"o is

Harris ergfodic. So the ergfodic theorem implies that, no matter what the distribution of

the starting value, Po,
m-1

j=0
is a strongly consistent estimator of E [ f(n) | y] ; that is, fm, E [f ( ) | y] almost surelyv

as m 00o. As defined in C'!s Ilter 2, we ;?i- that there is a CLT for fm if there exists a

0.2 E (0, 00) such that, as m 00o,

ImT, E [ f() | y]) N(0, 02) aS ,n ix OO .-1)

As explained in OsI Ilpter 1, establishing the central limit theorem for fm is crucial to make

honest statistical inference based on {py }Ro. We know that one way to ensure CLT in

(3-1) is by establishing geometric ergodicity of {@}Ro. In this chapter, we prove that the
Markov chains underlying the AC and PX-DA algorithms both converge at a geometric

rate which implies that the CLT in (3 1) holds for every fe L2(7i(lj Iy)); that is, for

every f such that fy, f2(P)xr(PIy)dp < 00. We also establish that PX-DA is theoretically
more efficient than AC in the sense that the .I-i-mptotic variance in the CLT under the

PX-DA algorithm is no larger than that under the AC algorithm. Regenerative methods

are used to construct a simple, consistent estimator of the .I-i-ing d o)tic variance in the CLT.

As an illustration, we apply our results to van Dyk and Meng's [2001] lupus data. In this

particular example, the estimated .I-i-inidllicl~ relative efficiency of the PX-DA algorithm

with respect to the AC algorithm is about 65. Hence, even though the AC and PX-DA

algorithms are essentially equivalent in terms of computational complexity, huge gains in

efficiency are possible by using PX-DA.

The remainder of this chapter is organized as follows. Results that the AC and

PX-DA algorithms are geometrically ergodic appear in Sections 3.2 and 3.3, respectively.

In Section 3.4 we derive results that allow for the consistent estimation of .li~!!llh d ic

variances via regenerative simulation.

3.2 Geometric Convergence and CLTs for the AC Algorithm

We begin with a brief derivation of the AC algorithm. Let RW = (0, 00), R_

(-oo, 0], z = (zi, .., z,)T E R" and let #(v; p, x2) denote the N(p, x2) density function

evaluated at the point ve R Consider the function from RW"x RW" R W given by

x(P,v z )=Is z) )ys a zeIo v)(zi; xf f )

where, as usual, IA (. iS the indicator function of the set A. Note that

i= 1

and hence, xr(P, z | y) can be viewed as a joint density in (p, z) whose marginal is the

target density xr(P | y). This joint density is usually motivated as follows. Let Z1, ..., Z,

be independent random variables with Zi ~ N(xTP, 1). If we define = Ipg (Zi), then

Yi, .. ,Y, are independent Bernoulli random variables with Pr(~ = 1) = # (xT f). The Zi's

can therefore be thought of as latent variables (or missing data) and xr(P, z | y) represents

the posterior density of (p, z) given y under a flat prior on p. The AC algorithm is simply

a data augmentation algorithm (or two-variable Gibbs sampler) based on the joint density

xr(P, z | y). Indeed, a straightforward calculation reveals that

| z y ~ N ( (X'X)X )-1

and conditional on (P y), Z1, .. ,Z, are independent with

Zi | 79, y ~ TN(.r {/, 1, yi).

If we denote the current state of the Markov chain as /9 and the next state as /9', then

the Markov transition density of the AC algorithm is given by

Note that k(P | 79) xr(P I| y) = k(P I | 7') xr(P | y) for all /3, /' E RIW'; i.e., k(P | 79) is reversible

with respect to r(fi | y). It follows ininediately that the posterior density is the invariant

density for the Markov chain, or, in symbols,

Let K(-, ) denote the Markov transition function corresponding to the AC algorithm;

that is, for /9 E RIW and a measurable set A,4

The corresponding m-step Markov transition function is denoted by K'"(ft,4). We now

show that the Markov chain driven by k(fi' | 79) is Harris ergodic.

Let p- denote Lebesgue measure on RIW. Several nice properties follow front the fact

that K(fi, -) has a (strictly positive) density with respect to p. Indeed, if 7t(,) > 0, then

K((P, A) > 0 for all /9 E RIW'; i.e., it is possible to get front any point 79 E RIW to the set ,4 in

one step. This implies that the AC algorithm is ys-irreducible and aperiodic.

In order to establish Harris recurrence, we must introduce the notion of harmonic

functions. A function b : RIW" R is called harmonic for K if h(fi) = (Kh)(fi) for all

/9 E RI>. One method of establishing Harris recurrence is to show that every bounded

harmonic function is constant [34, Theorem 3.8]. Suppose b is a bounded, harmonic

function. Since the AC algorithm is (-irreducible and has an invariant probability

distribution xr(P | y), it is recurrent, which in turn implies that & is constant ~-a.e. [34,

Proposition 3.13]. Thus, there exists a set NV with p(NV) = 0 such that h( ) = c for all

pe N Now, for any pe R P~, we have

which implies that h c. It follows that the AC algorithm is Harris recurrent.

We have now shown that the Markov chain corresponding to the AC algorithm is

Harris ergodic and thus from C'!s Ilter 2 it follows that ergodic theorem holds for it. The

following theorem is the main result of this section.

Theorem 1. The M~arkov chain on RW with transition 1:.: l 0 k (P' | P) (that is, the

Markov chain underlying the AC rlly.>rithm) is geomen,... ellol ergodic.

Proof. We will show that the AC algorithm satisfies the hypothesis of Proposition 1.

We have shown that AC algorithm is p-irreducible and periodic, where p denote the

Lebesgue measure on RP". So, if is a maximal irreducibility measure for the Markov

chain underlying the AC algorithm, then > p. Conversely, if p(A) = 0, then Km(P, A)=

0 for all pe R P? and all me N which implies that ~(A) = 0 and it follows that p > ~.

Hence, pa -. Since the support of p obviously has non-empty interior, it follows that the

support of a maximal irreducibility measure for the AC algorithm has non-empty interior.

We now demonstrate that the Markov chain associated with the AC algorithm is a

Feller chain. Let P and O denote a point and an open set in R ", respectively. Assume

that {#1}"7,, is a deterministicc) sequence in RP~ with p& / such that pt i as

m c o. Two applications of Fatou's Lemma in conjunction with the fact that xr~z|4, y) is

continuous in yield

lim inf K (PA,O) > lim inf k (p'| m ~) dp

= lim inf x( R* | lnz ) z
J to mm

> x~fi' z, y) int inf ~rx~a|7,aU

=K(P O) ,

and hence K(-, ) is a lower-senticontinuous function. Hence, the Markov chain corresponding

to the AC algorithm is a Feller chain.

We apply Proposition 1 with drift function V(fi) = (X/S)T(X/S). Recall that X is
assumed to have full colunin rank, p, and hence XTX is positive definite. Thus, for each

,* > 0, the set

P liE RI'? : V <) I 19=( E RIW : pTXTXp IS <

is compact so the function V is unbounded off compact sets. Now, note that

(KV) (p> ~ i)kf'7)p

= EE [V(:i') r-, ] i?-Y y

where, as the notation elo----- -r- the expectations in the last two lines are with respect

to the conditional densities xr(f' | x, y) and xr~x | 7, y). Recall that xr(f' | x, y) is a

p-dintensional normal density and xr~x | 7, y) is a product of truncated nornials. Evaluating
the inside expectation, we have

E [V(P ') zy] = E [(P ') X X/S' z,y]

=tr(X X(X X)-1) +: XX(X X)-1(X X) (X X)-1X :

=p + X X(X X)-1X :

< p+X z,

where tr(-) denotes trace of a matrix and the inequality follows from the fact that

z I-X X(XX -1TX z>0

for all z E R"n. We now have that

EE ;[V(I') z, ] ;), y < Ep +z z y =p+ E [z: | ,y].
i= 1

Standard results for the truncated normal distribution [19] imply that if U ~ TN((, 1, 1)

then,

E(U2) ~ 2

where #(-) with only a single argument denotes the standard normal density function; that

is, #(v) is equivalent to ~(v; 0, 1). Similarly, if U ~ TN((, 1, 0) then,

E(U2) ~ 2

It follows that

1 +( x pj ) ( if y = 0 .
A more compact way of expressing this is as follows:

SLiE [zf2 | ,y]= 1+(re-i) 4)2 ,/,Tn (3-2)

where I, is defined in the Appendix. Hence, we have

(KVT)(i) = E E[V;(P') zly] i,y) i= i

Recall that the goal is to show that (KV)(P) < AV(P) + L for all PR E W. It follows from

(3-3) that (KV)(0) < p + n. We now concentrate on P eRI \ {0}.
We begin by constructing a partition of the set RW \ {0} using the a hyperplanes

defined by wTP = 0. For a positive integer m, define Nm = {1, 2,..., m}. Let

Al, A2,..., Aan denote all the subsets of No, and, for each j E IT_ define a corresponding

subset of p-dimensional Euclidean space as follows:

Sj = {p3E R"\{0} :wfgT~<0fo rall i EAjand wfg7P >0fo raill i EAj}

where Aj denotes the complement of Aj, that is, Aj = N, \ Aj. Note that

the Sj are disjoint,

U zlSj = Rw \ {0}, and
some of the Sj may be empty.

We now show that if Sj is nonempty, then so are Aj and Aj. Suppose that Sj / 0 and

fix p E Sj. Since the conditions of Proposition 5 are in force, there exist strictly positive

constants {ai }", such that

Therefore ,

aimT + a~lTP + ---+ I, ,tiP = 0. (3-4)

The matrix X has full column rank p, and hence 0 < pTXTXp = CE (xTP)2

E = /ET 72. Thus, there exists an ie N such that wTP / 0 and, since all the ai are
strictly positive, (3-4) implies that there must also exist an i' / i such that I, 4 n r

have opposite signs. Thus, Aj and Aj are both nonempty. Now define C = { je E : T

0}. For each j E C, define
/ieA T n2 ieA.~ zT n2

and

xj = sup Ry (P) E [0, 1].

In the following calculation, we will utilize a couple of facts concerning the so-called

Mill's ratio. First, when n > 0, a (u)/(1 #(u)) > u2 [11, p.175]. Also, it is clear that if

we define

M=sup
ue(-oo,o] 1 (u) '(U
then M~E(0, 00).

Fix j E C. It follows from (3-3) and the results concerning Mills ratio that for all

SE Sj, we have

i=1

< p+ n + (,, )2

1- P)(wP)
ieA ~(~i

iEAj

(KV) (0)

-` (w 4p)2
iEj

(I,.TP)~(WTP)
1-~(WTP)

ie:

iEAj

< p+ n+ (w, P)2 + nM
i=1

=p + ~n(M + 1) + (w 4Tij)2

= xvp +nM+ 1)+R()(L)

where L := p + n(M~ + 1). Therefore, since s~cS,

(K V) (P) < AV (P)+ L ,

where

A := max Xj
jec

Hence, it suffices to show that As < 1 for all j E C.

Again, fix j EC and note that for le R ,, Rj(10)

Rj(P) which means that Rj(P)

depends on P only through p's direction and not on its distance from the origin. Thus,

xj = sup Ry (P = sup Ry (P < sup Ry (P) ,
P6S3 PES3 PES:*"

RIW \ {0}, it follows that

where

Sf* = {p E RP : ||4|| = 1 and wffT < 0 for all i E Aj and wff'i > 0 for all ie Ay } ,

and

Sf* =I {p n:|4|=1adwf o l lA n wf>0frali s

Now sice j*is a compact set in RP" and Rj(P) is a continuous function on Sj**, we know

that

sup R ( ) = Ry ( ) for some p E Sj**

Assume that p E Sj* is sulch that R,( ) =1, that is,

-2 -2

This implies that CE ,l(ll'n O) Again, there exist strictly positive constants

al, a2, to Such that

a~wim + a~wp + --+ a,wT =0o.
But we already, know, that wf#T = 0 for ll i e Aj, and hence it must be the case that

Howecver, w{~i < 0 for a~ll i eA, as a Sf*. This combined with th~e fact thait as a~re all
strictly, positiven shows that w{,T = for all i E Aj. Hence, we have identified a nonzero

such that

ITr = 0 for all ie N,

But this contradicts the fact that W has full column rank. Therefore, we have established

that

sup Ry (P) <1 ,

which implies that As < 1. Therefore, A < 1 and the proof is complete.

Together with the results of Roberts and Rosenthal [41], Theorem 1 implies that

the AC algorithm has a CLT for every f e L2(i( l g. Ill Order to use this theory to

calculate standard errors, we require a consistent estimator of the .-i-mptotic variance, o.2

This topic will be addressed in Section 3.4. In the next section we show that geometric

ergodicity of the AC algorithm implies that of the PX-DA algorithm and that PX-DA is
at least as good as AC in terms of performance in the CLT.

3.3 Comparing the AC and PX-DA Algorithms

The Markov transition density of the PX-DA algorithm can be written as

k*(p'l | )=xp ',yRz z)>z|4 )d

where R(z, dz') is the Markov transition function induced by Step 2 of the algorithm

that takes z z = (gzi,..., gz,)T. It is straightforward to show that the Markov

chain driven by k* is Harris ergodic. Hobert and Marchev [17] provide results that

can be used to compare different data augmentation algorithms (in terms of efficiency

and convergence rate). In order to establish that their results are applicable in our

analysis of k*, we now show that R(z, dz') admits a certain "group representation." Let

"xz | y) = fy, xr(P, z | y) dp. A simple calculation reveals that

|XTX|-4 exp ( z (I H)z/2}
cy~ly (y 2x

where H = X(XTX)-1XT. Let G be the multiplicative group R,+ where group

composition is defined as multiplication; i.e., for gl, g2 E G, gl o g2 = 192. The identity
element is e = 1 and g-l /.Telf-armaueo su~g dg/g where dg

denot~es L~ebesgue: measure: onl R Let~ GJ act on the left of RW" through component-wise

multiplication; that is, if geG c andu z e R", then gz = (gzi,..., gz,). With the left group

action defined in this way, it is easy to see that Lebesgue measure on RW" is relatively left

invariant with multiplier X(9) = 9"; i.e.,

J~S R ("ll J Rild

for all geG c andu all int~egrable functions h : R's R I. (See ChI Ilpters 1 & 2 of Eaton [10]

for background on left group actions and multipliers.) Let Z denote the subset of RW" in

which x lives; i.e., Z is the Cartesian product of n half-lines (RW and RW_), where the ith

component is R,+ if yi = 1 and RW_ if yi = 0. Fix x E Z. It is easy to see that Step 2
of the PX-DA algorithm is equivalent to the transition x i gx where y is drawn from a
distribution:- on G having density function

X(9);g ( y)v4(d9) 9;z-ls(gz y)dyg (c'( H):) ZT- -g' (IHiz/2 d
.fe X(9)~rg ( y)v4(d9) .1 9;z-li(gx | y)dy 2(it-2)/2F(n/2) eg.

Furthermore, S, (g)xr(gx | y);4(dy) is positive for all x E Z and finite for almost all: E Z.

Consequently, we may now appeal to several of the results in Hohert and Marchev [17].

First, their Proposition :3 shows that R(x, dr') is reversible with respect to xr~x | y) and it

follows that k*(f3'|/3) is reversible with respect to xr(P3| y). We now use the fact that the

AC algorithm is geometrically ergodic to establish that the PX-DA algorithm enjoys this

property as well.

Corollary 1. The M~arkov chain on RP" with transition I7:. um / k* (f' | /3) (that ise. the

M~arkov chain underlying the PX-DA rlly~.:,thm) is geomtl,... ril;i ergodic.

Proof. Define

Let K and K'* denote the Mlarkiov operators on L (r(n3 y)) associated with the Markiov

chains underlying the AC and PX-DA algorithms, respectively [25, :32]. Denote the norms

of these operators by ||K|| and ||K*||. In general, a reversible, Harris ergodic Markov

chain is geometrically ergodic if and only if the norm of the associated Markov operator

is less than 1 [41, 44]. By Theorem 1, the AC algorithm is geometrically ergodic and

consequently ||K|| < 1. But LiU and Wu [27] show that ||K*|| < ||K|| [see also 17, Theorem

4] and hence ||K*|| < 1, which implies that the PX-DA algorithm is also geometrically

ergodic. O

We have now shown that the Markov chains underlying the AC and PX-DA

algorithms are both reversible and geometrically ergodic and hence both have CLTs

for all f e L2(r(p |)). We now use another result from Hobert and Marchev [1'7] to show

that the PX-DA algorithm is at least as efficient as the AC algorithm.

Theorem 2. Fix f e L2(P Iy).I a~,ka a~,k d6801 the UGnritRCeS in the CLT f~or

the AC and PX-DA rlly.., /thms,, i~ 1.: 0 .l;, then 0 < o)~,k* a~,k
Proof. The result follows immediately from Hobert and Marchev's [2008] Theorem 4. O

In order to put our theoretical results to use in practice to compute valid .I-noi- nd l'lc

standard errors, we require a consistent estimator of the .I-i-mptotic variance and this is

the subject of the next section.

3.4 Consistent Estimators of Asymptotic Variances via Regeneration

We begin with the AC algorithm. Instead of considering the Markov chain on RW

driven by k(P' | p), we consider the joint chain on RW"x RW" with Markov transition density

given by

The Markov chain dlefined by k, which we denote by {pyj, zy}Ro,1 has invariant dlensity

xr(P, z | y) and satisfies the usual regularity conditions. It may seem to the reader more

natural to ulse the Malrkov transition density k (z', p' | z, 4)= p'|zyxz| ,)

for the joint chain. We have discussed this issue in Remark 3. Of course, the marginal

chain {@}Rj"o has the Markov transition density k(P' | P) no matter which version of the

Markov transition density we choose for the joint chain. While this is obvious for the

chain corresponding k, it can be easily shown for k by considering two consecutive steps

of the joint chain. Let k(P'|4) be the Markov transition density of the {@},t"o chain
corresponding k. Suppose, two consecutive steps of the joint chain are (P, z) and (P', z').

Then ,

=. k(p'|z, P)k(z|4)dz

where the conditional densities in the third equality are obtained from the Markov

transition density, k, of the joint chain. The de-initializing arguments of Roberts and

Rosenthal [42] can be used to show that the joint chain {pj,zy},t"o inherits geometric

ergodicity from its marginal chain {pj },to Note that {#y, zy }=,o is the chain that is
actually simulated when the AC algorithm is run (we just ignore the zys).

Suppose we can find a function a : RW x RW" [0, 1], whose expectation with respect

to xr(P, z | y) is strictly positive, and a probability density d(P', z') on RW"x RW" such that for

all (P', z'), (p, z) ERI? x R"n, we have

k (P', z | 4 z) > S (r, z) d(P', z') (3-6)

This is called a minorization condition [22, 30, 43] and it can be used to introduce

regenerations into the Markov chain driven by k. These regenerations are the key to
constructing a simple, consistent estimator of the variance in the CLT. After explaining

exactly how this is done, we will identify s and d for both AC and PX-DA.

Equation (3-6) allows us to rewrite k as the following two-component mixture density

k (p', z' | 4, ) = s (p, z)d(P', z') + (1 a (#, z)) r (P' z' | 4, z) (3-7)

where r is the so-called residual density defined as

k (P',z' | 4, z) Stp,z)d(P', z')
1 s(P, z)

when s(P, z) < 1 (and defined arbitrarily when s(P, z) = 1). Instead of simulating

the Markov chain {pj,zy}Rj"o in the usual way that alternates between draws from

"xz | p, y) and xr(P | z, y), we could simulate the chain using the mixture representation

(3-7) as follows. Suppose the current state is (pj, zy) = (p, z). First, we draw 6j ~

Brcnoulli(s(P, z)). Then? if ri, = 1, wei draw (iy I:, zy) from~ dl, anld if by = 0, we draw~n

(pj 1, zy41) from the residual density. The (random) times at which by = 1 correspond

to regenerations in the sense that the process probabilistically restarts itself at the next

iteration. More specifically, suppose we start by drawing (Po, zo) ~ d. Then every

time by = 1, we have (pj 1, zy 1) ~ d so the process is, in effect, starting over again.

Furthermore, the "tours" taken by the chain in between these embedded regeneration

times are iid, which means that standard iid theory can be used to analyze the .I-i- pind icl~

behavior of ergodic averages, thereby circumventing the difficulties associated with

analyzing averages of dependent random variables. For more details and simple examples,

see Mykland et al. [33] and Hobert et al. [16].

In practice, we can even avoid having to draw from r (which can be problematic)

simply by doing things in a slightly different order. Indeed, given the current state

(pj, zy) = (P, z), we draw (pj 1, zy 1) in the usual way (that is, by drawing from x~z | p, y)

and xr(P | z, y)) after which we "fill in" a value for by by drawing from the conditional

distribution of 6j given (pj, z) and (pj 1, zy 1), which is just a Bernoulli distribution with

success probability given by

We now describe exactly how these supplemental Bernoulli draws are used to construct a

consistent, estima~tor of E [f(p) | y] a~s well as a con~sistentt estim~a~tor of the correspon~ding

.I-imph!lli(c variance.

Suppose the Markov chain is to be run for R regenerations (or tours); that is, we

start by drawing (Po, zo) ~ d and we stop the simulation the Rth time that a by = 1.

Let 0 = -ro < -rTi r~ 72 < R be the (random) regeneration times; that is,

-te = min {j > -re-1 : by _l = 1} for te { 1, 2, .. ., R}. The total length of the simulation, -rR,

is random. Let NI~, N2.., ** R be the (random) lengths of the tours; i.e., Nst = -rt T-rt1

and define

N\ote that the (Ns;, St) pairs are iid. The strongly c~onsisten testimnator of E [ f (P) | y] is

S1

j=0

whee 3= R 2=1 Stan N= 2<1 N. Because the Markov chain driven by k is

geometrically ergodic, the results in Hobert et al. [16] are applicable and imply that, as

long a~s there exists anl au > 0 such that E[| f(P)|2+ | y] < 00, then1

d fr-E~f0)|y N(072 9)

as R c o. (Note that the requirement of a finite 2 + a~ moment is a bit stronger than the

second moment condition discussed earlier.) The main benefit of using regeneration is the

existence of a simple, consistent estimator of y2, Which takes the form

2 i~ t=1St i

Remark 1. The CLT in (3-9) is lkib~l;, different from the CLT discussed earlier, which

takes the form

Hobert et al. [16] explain that the two CLTs are retlated by the equation 2 =" E[s(P,z,) | y]O.2

Remark 2. A further ral.. tr,:y.: of using regeneration to calculate standard errors is that

the starting distribution is prescribed to be d(P,zx) so that burn-in is a non-issue.

We now derive a minorization condition for the AC algorithm using the "distinguished

In .11.1 technique introduced in Mykland et al1. [33]. First, note t~hat k (p', z' | 4,z) does not,

depend on P and as a consequence, neither will our function s. Fix a distinguished point

z, E R"n and let D be an p-dimensional hyper-rectangle defined by D = D x x D,

where Di = [ci, di] and ci < di for all i = 1, 2, .. ,p. Now note that

=s(z) d( ', z')

where

s (z)= E inf and d (P', z') = 1 rx (' | p', y) x (P' | ze, y) ID( ') (3-10)

and

J~~,( RP""( J.vi (l~"~( RnJD
Clearly, d(P', z') is a probability density on RW x RW". All that is required to apply the

regenerative method described above is the ability to draw from the density d (to start

the simulation) and the ability to calculate fl in (3-8). Making a draw from d( ', z') can

be done sequentially by first drawing P' from the truncated density E-1~(P / X* Y) D P)

(which does not require the value of E) and then drawing z' from xr~z' | P', y).

-1X z, I

exp zX(XX-Xz

exp z X(XTX)-] X z

We now provide a closed form expression for s(z), which in turn will give a closed

form expression for the success probability r7 First,

x(4 | z, y) = 1 exp
(2x)9|XTX|-

(z)) .

21

li(z)) XTX (4

where P(z)

S(z)

(XTX)- XTz Thus,

x(P | z, y)
E inf

E inf'
06Dexp (z-,'~x(,) XT z) 0XTX (z,)]

exp'

z' X(X7X)

z,) Xp)

exp c gIi =]

(tii )

(ti) + d Iti p

exp

SzT X(XT X)

'X1X

i~onf exp(z

where tT = (z z,)TX. Therefore, the success probability rl in (3-8) becomes

=~j mf) IDp~l j+1)
s#xD d5 ~, xjy) j1j1 +1 9

= mf ID j+1i

~(~lpj~ex)~pj~ ct Is (@ ) y I

exp -zXXX-Xz
( )x Xp I

ex

- -~~~

j+1)

where ti)T 1 z -z)TX and py 1,4 is the ith element of the vector 4 ~. Note that, rl

depends only on pja1 and zy. Also, note that, E is not required to calculate rl.

Notice that there is a chance for regeneration only when the P component enters the

p-dimensional rectangle D. This so__~-1-;- making D large. However, increasing D too

much will lead to very small values of rl. Hence, there is a trade-off between the size of D

and the magnitude of the success probability, rl.

Modifying a computer program that runs the AC algorithm so that it simulates

the regenerative process is quite simple. Since code for simulating from xr( '|z, y) and

xr~z' | p', y) is already available, it is straightforward to write code to simulate from d. All

that remains is a small amount of code to calculate rl and compare it to a Uniform(0, 1)

after each iteration of the AC algorithm.

Remark 3. A minorization condition can also be obtained for k using the "distinguished

1p~...ni" technique. However, in this case, the z component has to enter an n-dimensional

i ),l

exp -T zf X(TX)~-X zy

exp [i I =) 1 I /(j) ) 1 6"ID j+

r LItri:ll~. before a regeneration is possible. In most applications, a is much Irlary ? than p,

and when n is Iary,l the <.-?.;?7.17~l;, that all n components of z -:::l,,tll~r, .... ;, le; enter their

assigned interval is ini'~..a ll;i so small that the rll' .-r:thm is of no practical use. Moreover,
as mentioned above, this problem cannot be solved .: cl~;, by I,,;;.;. .9l the intervals 'ary ,I .

Regeneration can also be used in conjunction with the PX-DA algorithm. Indeed, we
now show that a simple modification of our minorization condition for the AC algorithm

yields a minorization condition for the PX-DA algorithm. The Markov transition density
of the PX-DA algorithm can be rewritten as

k*(p' | 4)=xp z yhg|zxz| ,y gd

where h(g | z) is the density in (3-5). As before, instead of working directly with k*, we

consider the joint chain on RW"x RW" x RW with Markov transition density given by

k*(p', (z',g') P, (z, g)) = [h(g' | ')(z' | ', y)] x(P |gzy).

Let {#y, (zy, gj)} o denote the Markiov chain corresponding to k*. Since T(4 | y) is the
invariant density for k*(P'|4), we have

=( x)p | gz~~iy)L [hg| i~z|y) g z

and from this it follows that xr(P | z,: y) [h(g | z)xr(z | )] is the invariant density for

{pj, (zy, g))}Ro. As before, considering two consecutive steps of the joint chain, it
can be shown that the marginal chain {pjy}Ro has Markov transition density k*. It is
straightforward to show that the joint chain is Harris ergodic, and, as before, the chain

associated with k* inherits geometric ergodicity from its marginal chain {Pjy}Ro. We can

get a minorization condition for k* as follows

k*(p, (', g) |4, (, g) =[h.(g' |')7i(z' | /j', y)] (p' | gz, y)

cs( Is (t ) did Ig (t )

exp -~ zTX(XX-Xz

s(gjzy) d*(,By 1, (zy 1, gj 1))

.~ x(6gzyy (,Bl |zej, y)

expi gzX (XTX)-1X z p

exp z X (X X)-1X z,

exp g3 zX(X)Xz

=exp= ct *()Ipg (t () +dit*j Ip_ (t*j ) t ,y4ID;j1

j+1>

where t*C F) = (yzi z)7X. TheCoremI 2 states that the .l-i Intl'l' d ic va~rian~ce in? th~e CLTi

for the PX-DA algorithm is no larger than that for the AC algorithm, i.e, O < o-f,k*I

o-f~k < OO for all f e L2 (7(,6 |y)). However, we know from Remark 1 that the regenerative
method is based on a slightly different CLT whose .I-i-inidllicl~ variance has an extra factor

involving the small function, namely E(s() | y), from the minorization condition. Although

the small fumetions in the two minorization conditions that we derived for the AC and

PX-DA algorithms are slightly different, E(s(.) | y) remains exactly the same as shown

>l~ o( in x(,6 |)( gz, y) hg|z)xz|,,y) (6|zyID;

=s(gz)d*(,', (z',g'/)),

where the function s(-) is as defined before and

d* (,', (z', g')) = [h(g' | z')r(z' |,6', y)]x(,6' ze,y/)I7D ;6)

where E is alSo the same as before. Hence, for the PX-DA algorithm the function rl

becomes

i1

below

=~~g sIz~~ | y)z|y)1

i= 1
=~~~2 cos sz) '(I H)z') g"2- e-z' (I-H)z'/2

x exp, z' (I H)z'/2g2)C --n R {} i R {} i

1+ (z') (I H)7( z')l~21J (I~T(1~~X
2(n-2)/2F(n/2) (Z' g" ex (- '(I- )z/g2 /

where z-' = gz. Clonseque~ntly, if E[| f~(P)|2+ < OO FOT Some a? > 0 and if r ,k and

qj,k* denote the variances in the regenerative CLT for the AC: and PX-DA algorithms,

respectively, then 0 < yf~k* I ,k <00. Hence, PX-DA remains more efficient than AC in
the regenerative context.

We end this section with an illustration of our results using van Dyk and Meng's

[2001] lupus data, which consists of triples (yi, Xl, Xi2), i = 1,... ,55, where xsi and Zi2

are covariates indicating the levels of certain antibodies in the ith individual and yi is an

indicator for latent membranous lupus nephritis (1 for presence and 0 for absence). van

Dyk and Meng [51] considered the model

with a flat prior on p. We used a linear program (that is described in the Appendix) to

verify that C'I. in and Shao's [2000] necessary and sufficient conditions for propriety are

satisfied in this case.

In order to implement the regenerative method, we had to choose the distinguished

point z, as well as the sets [ci, di]. We ran the PX-DA algorithm for an initial 20,000

iterations starting from the maximum likelihood estimate of P given by P = (-1.778, 4.374, 2.428).

We took the distinguished point to be the average value of z over this initial run. For

is { 0, 1, 2}, let pi and as denote the sample mean and sample standard deviation of the

#is over this initial run. Wle set De [iA 0.09 si, A+ 0.09 s,]. (The factor 0.09 s, was

chosen by trial and error.)

While generating Gamma and multivariate normal random variables is straightforward,

we need an efficient algorithm for generating truncated normal random variables. We

used the accept-reject algorithm of Robert [39] to generate one-sided truncated normal

random variables. We ran AC and PX-DA for R = 100 regenerations each. This took

1,223,576 iterations for AC and 1,256,677 iterations for PX-DA. We used the simulations

to estimate the posterior expectations of the regression parameters and the results are

shown in Table 3-1. (Results in C'I. i. and Shao [6] imply that there exists a~ > 0 such that

E [|py |2+" ly < OO for je {0(, 1, 2}).) It is strikiing that the estimated .I-i-ing~l .i ic variances
for the AC algorithm are all at least 65 times as '609.-~ as the corresponding values for

the PX-DA algorithm. These estimates -II__- -r that, in this particular example, the AC

algorithm requires about 65 times as many iterations as the PX-DA algorithm to achieve

the same level of precision. (We actually repeated the entire experiment seven times and

th~e estimates of 7 ,k 7 ,k* ran~ged betweenl 40 anld 145.)

Table 3-1. Results based on R = 100 regenerations

AC Algorithm PX-DA Algorithm
Parameter estimnate s.e. (~fi;fk leStimnate s.e. (;jf;k* ,k
Po -3.060 0.097 -3.018 0.012 6;6.6;
P1 7.005 0.190 6.916 0.023 66.9
2a 4.037 0.121 3.982 0.015 63.1

CHAPTER 4
BAYESIAN MULTIVARIATE REGRESSION

4.1 Introduction

Suppose Yi, Y2, are d-dimensional random vectors satisfying the linear

regression model

where p is the k x d matrix of unknown regression coefficients, xi's are k x 1 vectors of

known explanatory variables and we assume that, conditional on the positive definite

matrix E, the d-variate error vectorS E1, ,E are independently and identically

distributed with common density

fH E) = exp E E-1E d1H(5) (4-2)

where H(-) is the distribution function of a non-negative random variable. The density

fH is a multivariate scale mixture of normals and it belongs to the class of elliptically

symmetric distributions. The density fH can be made heavy-tailed by choosing H
appropriately. For example, when H is a Gamma(" ,\ )distribu~tion fulncition, fH becomes

the multivariate Student's t density with v > 0 degrees of freedom i.e., the density of El

becomes proportional to [1 +-FT- v"E 2-6 II !!v of the results in this chapter a~re

specific to the multivariate Student's t regression model.

We can rewrite the regression model in (4-1) as

y = Xp +E

where y = (yl, .., y,)T is the a x d matrix of observations, X = (xl, x2., )T iS the

a x k matrix of covariates and E = (E, .. En) iS the a x d matrix of error variables. The

likelihood function for the regression model in (4-1) is given by

We consider the standard noninformative prior on (P, E), i.e., we assume that xr( E) oc

|E| 2 The posterior density takes the following form

where c2 9) is the marginal density of y given by

d(d+1)
where W CR 2W is the set of d x d positive definite matrices. Fernandez and Steel [12]

proved that c2 9) < OO if and only if a > d + k. In section 4.2, we give an alternative

proof of the posterior propriety. A byproduct of our proof is a method of exact sampling

from xr(P, Ely) in the particular case when n is exactly d + k. Throughout this chapter

we assume that n > d + k. We also assume that the covariate matrix, X, is of full

column rank i.e., r (X) = k. The posterior density in (4-3) is intractable in the sense that

posterior expectations are not available in closed-form. Also, our experience shows that

it is difficult to develop a useful procedure for making i.i.d. draws from xr(P, Ely). In this

chapter we focus on MC1| C methods for exploring the posterior density in (4-3).

We develop a data augmentation (DA) algorithm for xr( Ely) in section 4.3.1. It

has been noticed in the literature that the standard DA algorithm often suffers from slow

convergence [51]. Empirical and theoretical studies have shown that alternative algorithms

that are modified versions of the standard DA algorithm such as the Haar PX-DA

algorithm and the ;;;.,97, l.:1r augmentation algorithm often provide huge improvement over

the standard DA algorithm (Liu and Wu [27], van Dyk and Meng [51], Roy and Hobert

[46], Hobert and Marchev [17]). In section 4.3.2, we develop the Haar PX-DA algorithm

for the posterior density in (4-3).

We then specialize to the case when the errors, Ei S, have a Student's t distribution
i.e., the mixing distribution, H(-), in (4-2) is a Gamma(, ") disotribu~tion fu~nction.W

prove that in this case, under certain conditions, both the DA and the Haar PX-DA

algorithms converge at a geometric rate. As mentioned in Chapter 2, the geometric

ergodicity of a Markov chain guarantees the existence of central limit theorem. Using

results from Hobert and Marchev [17] we also conclude that the Haar PX-DA algorithm is

at least as efficient as the data augmentation algorithm in the sense that the .-i-mptotic

variances in the central theorem under Haar PX-DA algorithm are never larger than those

under the DA algorithm. Some of these results are generalizations of results from van

Dyk and Meng [51] and Marchev and Hobert [28] who considered the special case where

there are no covariates in the regression model (4-1), i.e., X = (1, 1,...,1)T and H is
Gamma(" \ ) distribu~tion fuinction

The rest of this chapter is laid out as follows. In the next section, we prove the

propriety of the posterior distribution. In section 4.3, we describe the DA and the Haar

PX-DA algorithms. In the last section, we compare the two algorithms and prove that

both the algorithms converge at a geometric rate.

4.2 Proof of Posterior Propriety

In this section we address the propriety of the posterior distribution. In particular, we

prove that xr(P, Ely) is a proper density for almost all y if and only if a > d + k.

Theorem 3. Let y be Lebesgue measure on R d". The posterior I1:.:; 0 Hr(P, |y) is

proper for y- almost all y if and only if a > d + k.

Proof. We want to show that

for y- almost all y if and only if a > d + k. Recall that the likelihood function f(y|4, E)

is itself an integral with respect to the distribution function H(-). We now use this fact

to rewrite c2 9) aS aHOther integral which is defined on a larger space and is easier to

handle. In order to simplify notation, we assume that H(-) has a pdf h(-) (with respect

to Lebesgue measure on R,+). We introduce latent data q = (ql, q2, > ) Such that,

conditional on (i, E), { (y;-, qj)) n are iid pairs satisfying the relations

and qj|#,C E h(-).

If we denote the joint conditional density of y and q hy fly, q|/3, E), then by (4-1) and

(4-2) it follows that

I

f(U 9 /S, E)49

f(UO91j,, / )f(4jP9) / E4j

f(U /SE). (4-5)

So we get

JW JR

JW JR L JR" J

R" WRf(y q|/S, E)x(/3, E)d/S dEdq

where the last equality follows by Fubini s theorem. Note that, the integrand in the right

hand side of the above equation is given by

II
f~ ~ ~~~n (y, q|/3 EYj i ) exp
(2xr) z|E|lz =1

2 )=1

I
/(UO/S E)

Let Q be an ax n diagonal matrix with diagonal elements (- -,i~ -). Then,

j=1

j=1

j= 1

j=1

=tr E-p 0 -p p pyQ

(X Q- X)-1XTQ-ly. The following definition of the

Arnold [2, C'!s Ilter 17]

Yj)TC-I(PTXj

where, = (XTQ-1X)-1 and p =

matrix normal distribution is from

Definition 2. [] Suppose a~ is an mxr matrix;, E and T are mxm and rxr non-negative

7. I;,../.: matrices. We .rt;, that Z has a matrix: normal distribution with parameters a~,

and T if Z is an m x r random matrix: having moment generating function given by

and we write Z ~ Nm,r (a, E, T) In this case we have E(Z) = a~. Moreover, if E and T are

positive 7. It...:I~ matrices then Z has the following I. ,.-.:1/ function

1 1
.fz(z) =ep -t{ ( ) ( ) .
(2xr)mr/21 r/2 rm/2 2

Since r(X) = k, it follows that X Q-1X and hence R is a p.d. matrix. Thus,

(2r)9 | |0| 2I2I exp -
(2xr) 2 |E|?

Itr E -y -p || hq)E -
2

d+(n-k)+ 12

(2xr) 2 2 I = ( i
We now assume that n > d + k and will show that c2 9) < Xo. We f Tst show that when

n > d + k, y Q- y <"0- ft is a p.d. matrix with probability one (with respect to the

Lebesgue measure on R~d"). Notice that

yTQ-ly rgS-1q = Y -1Y YT -1X(X Q-1X)-1X Q-1Y

= g~ yi~x Q- I-Q ( X)-X'iX Q-Q- (4-7)

and since I Q-2X(XTQ- X)- X Q-2 is an idenipotent matrix, it implies that

yTQ-ly ,TS2-if is a positive senli-definite matrix. Now we prove that y Q-ly ,rg2-11
is a p.d. matrix by showing that |yTQ-ly 79S-1<| / 0 (with probability one). Let A be

the n x (d + k) augmented matrix (X : y). Then,

A'Q-1 A = Q X

X Q-1X X Q-1Y

y Q-1X Y Q-1Y

Therefore ,

|A'Q- A| = |X Q- X||Y Q-1 y -YTQ- X(XTQ- X) XTQ- Y|

= X Q-1X||y Q-l 79S-11

= |C-ll 7 -l 79S-19|. (4-8)

Since r(X) = k, we know that X Q-1X is a p.d. matrix and hence |0| > 0. Also since

n > d + k, the d + k columns of A are linearly independent with probability one because

the probability of n dimensional random column vectors of y lying in any linear subspace
of R's with dimension I n 1 is zero (with respect to the Lebesgue measure on R~dit

So, A"Q-lA is a p.d. matrix and hence |A"Q-lA| > 0. Then from (4-8) it follows that

|yTQ-ly p'O2-1p| > 0.

To integrate the expression in (4-6) with respect to E, we use the following definition

of Inverse Wishart distribution.

Definition 3. [29, p. 85] Let 8 be u p x p p.d. matrix;. Then for some m > p the p x p

random matrix: W is said to have a Inverse Wishart distribution with parameters m and 8

if the p.d.f of W1 (wi!th, respectt to Lebesgue measure on, E. I, restrictedl to th~e set wrh~ere

W > 0) is given by

m+p 1-1 -1

mp p(p-1) m ]/ .-i)
2 2 x 4 102 __ -,,,

f (W, m, 8)

and we write W ~ IW,(m, 8).

Hence if a '> d + k i.e., a k > d by the above definition of Inverse Wishart

distribution, we have

2~ ) ~ x r 1/ 0( ( k 1-i)) a

whn =d + k
If a=1 d + k using (4-8i) wegl getll \

4-9)

2 4
aca+ |A hq).( -0
I i1 4d~-

JW JR

Since h(-) is a probability density function, it follows that in the case n = d + k,

which, of course, is a finite number. So, we have proved that in the particular case when

n = d + k, the posterior distribution is proper with probability one. Then, an application

of Lemma 2 of Marchev and Hobert [28] shows that for y-almost all y the posterior is

proper for n '> d + k.

We will now show that the posterior distribution is improper when n < d + k. Let

L = yTQ-ly p'O2-1p. It is easy to see that I Q-2X(XTQ-1X)-1XTQ-2 is an

idemlpotent matrix wsithl tr I 1 -X(XT-1'X)-~1XTQ- = n- k.- We also kllno that
rankly) = d with probability one. Hence, from (4-7) it follows that if a < d + k, then L

is a p.s.d. matrix i.e., there exists a vector xo(f 0) such that Exo = 0. We use this fact to

show that the integral in (4-9) diverges when n < d + k.

Since E is a symmetric matrix, the Jacobian of the transformation Z = E-l is given

by J = |Z|-(d ) [29, p. 36]. Therefore, we have

|E|exp tr(E 2)d Z exp -trZ) Z
w2 2

The matrix Z is p.d. So, by Clo d.1 -l:y decomposition, Z can be uniquely represented as

Z = LLT, where L is a lower triangular matrix (1.t.m.) with positive diagonal elements.

Th~e Ja~cobian of th~e corresponding transformationl is given by | J| = 2" nl0 1 -i+ 1[29, p.

36]. Let L = (li, 12 d). Therefore,

|f, Z| exp tr (Z) dZ= 2 | Lrn-kd-1ep -t L 1-+d
1i 1
= 2 |Lt-kd-exp -tr 1 -+L
i=1i= 1

=~~~~~ 2 x 101 1-kiL
ad~ ~ i 1 --dl~ i=1 1=

where

U = { (Un1, U21,, i2, L, Udd) E H2 ii > 0, Vi = 1, d}.

Notice that the columns of L form a basis of R d. So there exists constants bl, b2,..., bd

with b, j 0 for some i such that xo = 2, bil Suppose i'= min {ie {1, 2, .. ., d} : bi j

0}. Now consider the transformation L = (li, 12 d) O = (ox, 02, Od) Where

oi = 14 V i / i' and oi/ = zo i.e., O = LA, where

1 0 --- 0 --- 0

0 1 --- 0 --- 0

A=
0 0 --- bi/ --- 0

0 0 --- bd --- 1

Then the Jacobian of the transformation is | |A| | = | bi |-d [29, p. 36]. Note that

14, = of, -e D bo and in particular lits, = oi/i/ because oi/ = 0 for i > i'. Since

foi/ = 0 we have

d d
exp i1 01 1 -k-idL
ifi 1" i= 1

exp o s bo r- iso -d
i/i i' i>i/ i>i' i=1

exp o foT + 1c o Fo1 oi O-k-idO

vi/ ii/ i i=1

where

V = {(v1 v21, U22, 31, Udd) ER : 2 > 0, Vi / i' and A~ I > 0}.

By Fubini's theorem we can rearrange the order of integration. Notice that oi/ does not

appear in the exponential term and the only term involving oi/i/ in the above integral is

oppk-i'~. Hence the above integral dliverges since

oppk-[ i R i'~i')I(bs >o 0) +Is os,)rIcbs, 0u) does, = 0

So, we have now proved that the posterior is proper for y- almost all y if and only if
n~d+k. O

As we mentioned in the introduction that Fernandez and Steel [12] gave a proof

of propriety of the posterior density xr(P, Ely). A byproduct of our alternative proof is
a method of exact sampling from xr(P, Ely) in the particular case when n is d + k. We
describe the method now.

Let xr(q, p, Ely) be the joint posterior density of (q, p, E) given by

Then from (4-5) it is easy to see that

/( "(q, P, Ely)dq = xr( Ely). (4-11)

So iid draws from xr( Ely) can be obtained by making i.i.d draws from xr(q, P, Ery) (and

then just ignore the q component). Draws from xr(q, p, Ely) can be made sequentially via

the following decomposition

In the special case when n = d + k, from (4-10) we know that xr(qly) = nL, h(qi). So, in

this case an exact draw from xr( Ely) can be made using the following three steps:

(i) Draw 41,q2,...q,, independently where qi ~ h(qi).

(ii) Draw E ~ IWd[n k, (yTQ-ly yTQ-1X(XTQ-1X)-1XTQ-ly)- ]

(iii) Draw PT ~ N~d~k NT -1X(XTQ-1X)-1, E, (XTQ-1X)-1)

Standard statistical packages like R (R Development Core Team [36]) have functions for

generating random matrices from the Inverse Wishart distribution. One way to generate

Z ~ Nm,r(p-, E, T) is to first independently draw Ze ,pT hr steihrwo

p- for i = 1,..., m. Then take
ZT

ZT
Z= E2

ZTn

4.3 The Algorithms

In this section, we develop the DA and the Haar PX-DA algorithms for the posterior

density (4-3). We first develop the DA algorithm in Section 4.3.1 using the latent data

q = (qi, q2,., q n) and the joint posterior density of xr(q, p, Ely). We then derive the Haar

PX-DA algorithm in section 4.3.2. In the special case, when the observations, yi's, are

assumed to be from a multivariate Student's t location-scale model, van Dyk and Meng

[51] developed the marginal augmentation algorithm, which is a modified version of the

standard data augmentation algorithm, for the density (4-3). Hobert and Marchev [17]

have shown that when the group structure exploited by Liu and Wu [27] exists, marginal

augmentation algorithm (with left-Haar measure for the working prior) is exactly same as

the Liu and Wu's [1999] Haar PX-DA algorithm. In section 4.3.2 we show that a similar

group structure can be established for analyzing the posterior density in (4-3) and so

marginal augmentation algorithm is the same as the Haar PX-DA algorithm in our case.

4.3.1 Data Augmentation

We now describe the basic data augmentation (DA) algorithm. The DA algorithm

simulates a Markov chain with Markov transition density

where xr(P, E1q, y) and xr(q|4, E, y) are the conditional densities obtained from the joint

density xr(q, p, Ely).

Note that, k(P, E|4', E')xr(P', E'|y) = k(P', E'|4, E)xr(P, Ely) for all (p, E), (P', E') E

E"'' x W, i.e., k(P, E|4', E') is reversible with respect to xr(P, Ely). It then immediately

follows that xr(P, Ely) is the invariant density for the DA algorithm, i.e.,

Since the Markov transition density, k(P, E|4', E') is strictly positive (with respect to the

Lebesgue measure on E"'' x W), using similar arguments (that we used to show the AC

algorithm is Harris ergodic) as in Chapter 3, it can be shown that the DA algorithm is

Harris ergfodic. Then from the discussion in C'!s Ilter 2, it follows that the ergfodic averages

based on the DA algorithm can be used to estimate posterior expectations.

A single iteration of the data augmentation algorithm first uses the current state

(P', E') to generate q from xr(q|4', E', y) and then draws the new state (p, E) from

xr(P, Elq, y). Simulating from xr(P, clq, y) can be done sequentially by first drawing

E from xr(E|q, y) and then drawing P from x ( | E, q, y). From section 4.2, we know

that conditionally P|E, q, y follows a Matrix Normal distribution and the conditional

distribution of Elq, y is an Inverse Wishart distribution. Conditional on (P, E, y) qi's are

independent with

gi|#F, E, y in hfin x (x -y)E-(x -y

In the particular case when h(-) is Gamma (" ) ine., when yi, y2, .. "r i" i nrTOrS~umed to

be observations from Multivariate Student's t regression, we get

ind v + d v + (P Xi yi)TE-1(PTxi yi)
2' 2

4.3.2 Haar PX-DA Algorithm

In this section we derive the Haar PX-DA algorithm using the left Haar measure as

described in Hobert and Marchev's [2008] Section 4.3. From Section 4.2, we know that

"(q y) c |X Q-] X-I |y Q- y- QT- X(X Q--X- X)- _, X Q y qhi)
i= 1

As in chapter 3, let G be the multiplicative group R,+ where group composition is defined

as multiplication. Again, the left Haar measure on G is vl(dg) = dg/g where dg denotes
then Lebesgue m measure on Rm .Lt G1 act on R" through component wise multiplication,

i.e. gq = (ggi, ,.i.; ). Then it is easy to see that Lebesgue, measures on Rn is relatively

left invariant with multiplier X(g) = g". In order to construct the Haar algorithm, we need

to verify that m(q) = faxr(.i;|
given y7 can also be written as
d n-k d

(ur~uXTQ-1X yTQ-ly y Q-1X(XTQ-1X) X Q-1 y ii2 4i4 q4hB

Therefore it follows that xr(l-i.;|<) = c3 =1, */*]~ ) Where c3 1S a COnStant that does not

depend on g. Hence,

m~~~q)~ = 3O )O = C3
i= 1 i= 1

In1 order to show that n(ql) < co, sup~pose~ x ~ -h( ); x > 0 and consider th~e standard

noninformative prior, 1/o-, for the scale parameter, o-. Then the corresponding posterior

distribution is proper since

/1 x da 1 "x
-h()hyy(y=-
o o- o- o- xo

10

Then by Lemma 2 of Marchev and Hobert [28I it follows" thatI fo =1o~)$<0 for~ ~ ~ ~ ~ ~ ~ ~~~~'\ al o x,-- .> n n>1snc ,, a be viewed as the posterior density of the scale parameter, o-, when the standard noninformative prior, 1/o-, is combined with the likelihood function of a iid observations,xl, x2., *,n, from the scale family -h( ). He~nc~e, it follows that mr(q) < 00. Consider the following univariate dlensity on RW_ e,(g) = lit) m(q) i= 1 Then one iteration of the Haar PX-DA algorithm, which is a modified version of the DA algorithm, consists of the following three steps: (i) Draw q ~ xr(q|S', E', y) (ii) Draw g ~ e,(g) and set q' = gq (iii) Draw p, E ~ xr(P, E1q', y) The Markov transition density of the PX-DA algorithm can be written as where R(q, dq') is the Markov transition density induced by the step 2, that takes q q' = gq. In the special case when we have multivariate Student's t data, it is easy to see that the density e,(g) is Gamma (n f-). So, in this particular case, the step 2 of the Haar PX-DA algorithm is simply a draw from a Gamma distribution. Our results in the next section are specific to multivariate Student's t regression. 4.4 Geometric Ergodicity of the Algorithms In this section we prove the geometric ergodicity of the DA and Haar PX-DA algorithms. In C'!s Ilter 2, we mentioned that the geometric ergodicity of a Markov chain can be proved by establishing a drift condition and an associated minorization condition. In this section, we show that these conditions can be established for the DA algforithm and the Haar PX-DA algforithm. We now prove used to establish the drift condition. Lemma 1. Sup~pose that P is a p.d. matrix: and that P - vector x. Then, the following lemma that will be XXT is a p.s.d. matrix: for some X"P- x < 1. Proof. Consider the matrix Calculating the determinant of the above matrix twice using the Schur complement of P and 1, we get the following identity |P|(1 x P- X) = |P xx |, i.e., |P XXT | x P- X = 1- |P| Since P is a p.d. matrix, |P| > 0. Similarly, since P-XXT is a p.s.d. matrix, |P--xxT| > 0. Then from the above identity it follows that X"P-1X < 1. O The following lemma establishes the drift inequality for the DA algorithm. Lemma 2. Let V(P, E) = CE (yi PTXi)TE-l(yi PTX,). Then for the DA dy..~ rithm we have n+d-k n+d-k E(V(P, E)| I', E') < V(P', E') + v.. v+d-2 v+d-2 Proof. Recall that the Markov transition density of the DA algorithm is JRY ~(p, clp', C') So we have E [E {V~(P, E)|q4,y}|O'I, E', y] G(V(P, C)IP', C') (4-12) To calculate the above conditional expectations we need the corresponding conditional distributions. The required conditional distributions, as derived in the previous sections, are the following PT | q, y"1dr(-T ~, Nd E~q y~ Ie "Q-y p and ind (V+dv+(Ti yi T 1(P~ T i q4|#, E, y ~0 =12.. 2 2 Starting with the innermost expectation, we have E(V(P, E)|E,q, y) i= 1 i=1i i=: 1i i=- 1~ 1 i= 1 To/ cacuat th bv xetton eueteflo in prpryfmtixnra distribution.i Wk o -1pul T), Wh aoere by V~ W,(,9, we meahefllwng thapety V f h as x dmnsonal noncentral Wishart distribution with m degrees of freedom, covariance matrix W and with noncentrality parameter 6 [2, chap 17]. In this case, E(V) = mW + 5. If 6 = 0, we wi that V has a central Wishart distribution and we write V ~ W,(m, 9). So, we have E [E {E(V(p, E)|E, q, y)|q, y) |', E', y] . E [ E- |E, q, y] dR + pEI-1p-1 and hence, E(V(P, E)|E, q, ) =yTC yi i= 1 i= 1 Ij-~ p x4+ x[da + pE- p ] x4 i= 1 i= 1 i= 1 p 1Xi) E 1(y To calculate the second level of expectation in (4-12), we use the fact that if X~ IW,(m, 8), then X-l ~ W,(m, 8) and E(X-1) mO [29, p. 85]. Thus, E [E(V(P, E)|E,q, y)|Iq, y i= 1 p- xi)+di X 2Xi i= 1 p x4)T E(E-ly |~) (y n l xSi) + dC xf axs. i= 1 (4-13) p 1x ) (y Q- y p'O- p) (y Now, yTQ ly-p p ~T yTQ y p p S-1 2-T2pl p1 q ysy + p (X Q- X)p 2p 1X Q ly n i= 1 n i= 1 a~ ( q ys- p x )(ys i= 1 p 1X ) . (yi p 1X ) (y Qly pl p) ( p x j=1 (Yi I-1TXi) ( j= 1 4i 4i (ys p x ). T p xy(yy- pxj) Since we assume that n '> d + k, from Section 4.2, we know that yTQ-ly p'O2- p is poitve efiit matrix,,c:, withcl proablity:l c 1. So pn (y~ p xj)(yj p'xj) = yi Q-' y)-p 0- p11 1 is a p~d. matrix w-ith probability 1. Also, it is straigh~tforward to see is a positive semi definite matrix. So, an application of Lemma 1 yields (yi p Txi) (y Q-1 y p 0- )- (yi p x ) I -. (4-14) Since we assume that r(X) = k, it implies that X Q- X is a p.d. matrix and obviously xy/ x.:IT is a p~s~d matr~ix. So ano0ther7 application? of ~Lemma. 1 give~s xfaxe8 = x (X QolX- X) x j=1 < (4-15) Therefore, we get E(V( n, E)|q, y) < (n + d k) 1 i= 1 Now, recall that ind 1 V + d v + (Txi yi)TE-1(PT4 x -i ys) , q4|#, E, y ~ 0 =12.. 2 2 So, using the fact that if w: ~ Gamma(a, b) then E( ) = finally, we havei E(V(P, E)|#', E') 5(s-# i E)-(s-# i i= 1 which proves the lemma. O The following lemma establishes an associated minorization condition. v+ (p' x, yi) (E')- ( Xi -- yi)\ v + (P' xi yi) (E')- ( X, yi) Lemma 3. Fix: 1 > 0 and let S {((, E) : V(P, E) < 1}. Then the M~arkov transition 1, .:1;i of the DA rlly.>rithm k(P, E|4', E') .el.:. the following minorization condition k(P, E|4', E') > ed(P, E) V(P', E') ES where the 7. ;. -.1 / d (P, E) is given by at~ So gtd9(i ) d r and e = ( fo g(t)dt)". The function g(-) is given by i/t (v+d v vd v+1 9() F 2 2' (,* whAere q* = iSlogl 1+ andr I(a~blx) denotes the Gamlrmajab) I r.' .1/ evaluated at the point x. Proof. For i = 1, 2,. ., n, define p'x,) (E)- (ys PTXe) < 1}~ Si = {((, E) : (ys Clearly, S C Si for i 1, 2,. ., n. Recall that, xr(q|0, E, y) is product of a Gamma densities. So, Yi ) (E')- ( X, mnf r(q|S',E', y) (P',C')ES Then by Hobert's [2001] Lemma 1, it straightforwardly follows that (q|O', E', y) > gii)ca Vci', E') t S. i= 1 V + (P' Xi .P,')~= v + d i= 1 i=1 (P',C')ES 2 ' Hence, the proof follows because Fr-om results stated in chapter 2 it then follows that the DA algorithm is geometrically ergodic as long as Ithe coefficient of V(B', E') in Lemmnra 2, i.e., "- is strictlly less than 1. We state this in the following theorem. Theorem 4. The DA ,Ily.>rithmn is ~i..- tre. I :. .rli ergodic if 0 < < 1 i.ec., a2 < v+k-2. Hobert and Marchev's [2008] Proposition 6 shows that the Haar PX-DA algorithm is at least as efficient (in efficiency ordering) as the DA algorithm. Using similar arguments as in Corollary 1, we can show that geometric ergodicity of the DA algorithm implies that of the Haar PX-DA algorithm. Hence we have the following corollary. Corollary 2. The Haar PX-DA rlly. rithm is at least as efficient as the DA rlly.>rithm. Also, the Haar PX-DA rlly.>rithm is geomen,... ril;i ergodic if a < v + k 2. Remark 4. Our result in Lemma 2 matches with M~archev and Hobert's [2004]l Theorem 1' in the case when k = 1. We also can prove the geometric ergodicity of the Haar PX-DA algorithm by directly establishing a drift and minorization condition for it. We actually can use the same drift function V(P, E) = CE (yi PTXi)TE-l(yi PTX,) to establish a drift condition for the Haar PX-DA algorithm. Lemma 4. Let V(P, E) = CE =(yi PTXi)TE-l(yi PTX,). Then for the Haar PX-DA rlhi ~.:athm we have (n +d k)(v +d)(n 1) av(n +d k) n(n 1)(n + d k)v(v + d) E ( V (, E) | ', E')I (nv 2)(v +d 2) nv 2 (nv 2)(v + d 2) Proof. Recall that the Markov transition density of the Haar PX-DA algorithm is - yi) E-1(P X, - So we have V(#, E)k(p, E|4', E')dpdE JW JR E [E { V(#, E)|q, y) | ', E', y] E(V(P, E)|# ', E') From Section 4.3.2 we know that q' gq where giq ~ Gamma (n ~-). We substitute q by q' in (4-13) and then straightforward algebra shows that (vi p xi) + xOx i= 1 n k) 9 p 1Xi) (y Qly pl p) (y E(V(P, E)|q1, y) E(V(P, E)|q, y) = (n k) (ys l i = 1 i i= 1 p'1Xi) (YTQ- y p'1Xi) (YTQ- y p'O- p)- (y p'O- p)- (y p- xs) + di x Ox i= 1 From (4-14) and (4-15) it then follows that v~n d -k) "1 nu 2 q4 i= 1 E(V(P, E)|q, y) v(n +d k) g av 2 q::4j~ i= 1 j i Since conditional on (P, E, y), qi's are independently distributed with in s v +d V + (PTxi q4|#, E, y ~ 1,2,...,n, we have E(V(P, E)|4', E') v(nn +- d k) n+ gy j~~--> i= 1 j i v(n + d k) [ v + -vd V + (P'TXi - av-2 + 2 i=1si v~ V+ (Pl'z Yi )T(E')-1(P'TXs -yj)TE'- (P'Txy -yj>Y) E q, y yi>)2 v(n + d k) I (n 1) (v + d) (' )CI-PTX nu 2 v(V+ d 2) - which proves the lemma. O Our minorization condition for the DA algorithm straightforwardly generalizes to a minorization condition for the PX-DA algorithm. Lemma 5. The M~arkov transition I7.* .:1/ of the Haar PX-DA rlly..,7:thm k(P, E|4', ') al. J. the following minorization condition where the 7. ,: -.1 / d (P, E) is given by and e, S and g () are as .I~ .,: .1 in lemma S. Together, Lemmas 4 and 5 prove the following theorem. Theorem 5.'lt The Har PX-DA rly. rithmr is geomenl~:...rl c/ ergodi if 0 < (n~'d-k vd n1 As a corollary of Theorem 4 (see Corollary 2) we know that the PX-DA algorithm is geomnetric~ally erg~odic if < 1. At first it might appear that Theoremr 5 is a better result than Corollary 2. But, we now show that it can never happen that both of the followingf inequalities hold together n + d-k (n d -k)(v d)(n -1) > 1 < 1. (4-16) v+ d- (nv -2)(v d -2) N\ote that (n d-)(v'd)(n-) n '+d-k if and only if (" ")( < 1 i.e. nr < 1 + "-2. So, if (nv- 2)(v+d- 2) v+d-2 (nv-2) d (4-16) holds then the following must holds n+d-k v-2 > 1; ,a< 1 (4-17) v+d-2 d which, in particular, we~ that v has to be '?N r;i than 2. Now (4-17) holds if and only if v-2 n~v+k-2andn<1+ But, clearly, the above two inequalities can't hold together. We, of course, would like to be able to wi that the DA algorithm ( or the Haar PX-DA algorithm) is geometrically ergodic for all (d, k, v, n)-tuples such that d > 1, k > 1, v > 0 and a > d + k. Note that the conditions in Theorem 4 and Theorem 5 are upshots of the particular drift function V(P, E) and the inequalities (4-14) and (4-15) that we used to prove the drift conditions. In our opinion, to make substantial improvement of Theorem 4 and Theorem 5, we either have to consider a different drift function or resort to a altogether different technique of proving geometric ergodicity other than establishing drift and minorization condition. Either of these two would require us to start from scratch. CHAPTER 5 SPECTRAL THEOREM AND ORDERING OF MARK(OV CHAINS This chapter is divided into two sections. In the first section we present a brief account of operator theory on Hilbert space. In particular, we define the spectral theorem for a bounded normal linear operator on Hilbert space. In the second section we show how these results from functional analysis are used in studying the theory of Markov chains. 5.1 Spectral Theory for Normal Operators We begin with the definition of a Hilbert space. Definition 4. A complex vector space H is called pair of vectors y and b in H is associatedd a complex number (g, h). called the inner product of g and b. .such that the followings hold: (g, h) =(h, g) (g + h, k) = (g, k) + (h, k) if h, k E H (a~g, h) = co(g, h) if 9, h EH and n~EC@ (h, h) > 0 for <<11 h EH and (h, h) = 0 only if h = 0. It can be .;-.le;, shown that the above inner product induces a norm on H I. I;,.. l by | |h | = (h, h) E If the r. tel. e:11 normed space is complete. it is called a Hilbert space. Let H he a Hilbert space over C and T : H H he a linear transformation, called a linear operator. The operator T is said to be bounded if there exists an At > 0 such that ||Th|| < Af||h|| for all h e H, where the norm, || ||, is as defined above. Let B(H) be the collection of all linear, bounded operators from H into H. For Te E (H), define the operator norm of T by i ~i||Th||ll htlhO It is easy to see that ||T|| = sup{||Th|| : h e H, ||h|| < 1} = sup{||Th|| : h e H, ||h|| = 1}. It can he shown that B(H) with the above norm is a Banach space (i.e., it is complete). If S, Te E (H), the composite operator ST E B(H) is defined by (ST)(h) = S(T(h)) h e H. In particular, powers of Te E (H) can he defined as To = I, the identity operator on H and T's = T,-1, for n = 1, 2, 3, .. We can easily verify the inequality || ST|| < || S|| || T|| Now, we define the adjoint of an operator. Let Te E (H). Then using Riesz representation theorem it can he shown that there exists a unique T* E B(H) for which (Ty, h) = (g, T*h) for all g, h e H. The operator T* is called the adjoint of T [47, p. 311]. Some properties of adjoint operators are listed below: ||T*|| = ||T|| T** =T ||T*T|| = ||T||2 and if S also belongs to B(H) then (ST)* = T*S*. We now can define different types of operators. An operator Te E (H) is said to be normal if TT* = T*T, self-adjoint if T* = T, unitary if T*T = I = TT*, projection if T2 =T Clearly, self-adjoint and unitary operators are normal. Note that, if T is self-adjoint then (Th, h) = (h, Th) for all h e H. But, from Definition 4 we know that (h, Th) = (Th, h). So, for self-adjoint operator T, (Th, h) is real for all h e H. We now state a useful uniqueness theorem [47, p.310]. Theorem 6. If Te E (H) and if (Th, h) = 0 for every h E H. then T is the zero-operator. Remark 5. The above theorem would fail if the .scatler field were RW instead of C. For exravple. consider the mardrixr T = < t s As a corollary of the above theorem we get an alternative definition for a normal operator. Corollary 3. An operator T on H is a normal operator if and only if ||T*h|| = ||Th|| for every h E H. Proof. Suppose that || T*h || = || Th || for every h E H. Then || T*h1 || 2 h12 and (T*h, T*h) = (Th, Th). This yields (TT*h, h) = (T*Th, h) i.e., ((TT* T*T)h, h) all b in H. By the above theorem we get TT* = T*T, i.e., T is normal. Conversely, if T is known to be normal then we can write ((TT* T*T)h, h) all b and now following the above reasoning in the reverse direction we get ||T*h|| for every h E H. so = for 0 for = |Th|| O Suppose Te E (H). The operator T is invertible if there is an S E B(H) such that TS = I = ST. The operator S is called the inverse of T and we write S = T l. Suppose R(T) denotes the range of T, i.e., R(T) = { Th : h e H}. Then T is invertible if and only if R(T) = H and T is one-to-one. The spectrum, o-(T), of the operator Te E (H) is defined as follows o-(T) = {A : T AI is not invertible}. Thus Ae o (T) if and only if at least one of the following two statements is true (i) The range of T AI is not all of H i.e., T AI is not onto. (ii) The operator T AI is not one-to-one. If (ii) holds, A is said to be an 1.:I ,: al;,.- of T and in this case there exists & / 0 such that Th = Ah. We call h an eigenvector corresponding to the eigenvalue A. It is difficult to make a general statement about the spectrum of an operator. We define compact operator later in this section. We will see that we can ;?, quite a lot about the spectrum of a compact operator. The complement of o-(T) is called the resolvent set of T and is denoted by p(T), i.e., p(T) = C \o-(T). The proof of the following theorem can be found in text books on functional analysis [see e.g. 9, p. 83]. Theorem 7. If ||T || < 1, then (I T) is an invertible operator. Furthermore, n= o Corollary 4. Let A be a complex; number such that ||T || < |A|. Then from the above theorem it follows that T AI is an invertible operator and its inverse is noi Corollary 5. Suppose that T is an invertible operator and S is another bounded linear operator such that ||T S|| < ||T-1|| ' then S is an invertible operator. Proof. We have || T- S I = IT-(T S)I < IT- IIT SI < 1 by our hypothesis. Thus, by above theorem, I (I T-1S) = T-1S is an invertible operator. Let us denote the operator T-1S by D. Then S = TD and so D- T-l is the inverse of S showing that S is an invertible operator. O Remark 6. Cor .ll.r,, a tells us that a(T) is a bounded set for r,.;; bounded linear operator T because a(T) C {A E C: |A| < ||T||}. Theorem 8. For r ;, bounded linear operator T the set a(T) is a closed, bounded subset of C. Proof. We already know that a(T) is a bounded set. We only need to show that a(T) is a closed set. We will show that p(T) is an open set. Let A be any point in p(T), i.e, (T AI)-l is a bounded linear operator. Choose p such that |p 1- A| < ||(T AI)- ||-l and let T' = T AI, T" = T pl. Now T' is invertible and From Corollary 2, it follows that T" is an invertible operator. Thus, p E p(T) and in particular it follows that i.e., any A s p(T) has a neighborhood contained in p(T). Hence, p(T) is an open set. O It can also be proved that for Te E (H), H / {0}, the spectrum of T, o-(T), is not empty [47, p.253]. The spectral radius, rT, of T is defined as rr, = sup{|A| : A E o-(T)}. So, rT is the radius of the smallest disc (with center at 0) containing the spectrum of T. From Remark 6, we see that rT < ||T||. We now state the spectral radius theorem. Theorem 9. (Sp~ectral Radius Theorem) If Te E(H). the spectral radius. rT. of T r, = lim || T'sl || in f ||T'j The operator norm of a normal operator is same as its spectral radius. We state this as the following theorem. Theorem 10. Let Te E (H) be a normal operator. Then rT = ||T||. If T is self-adjoint then a(T) is a non-empty compact subset of RW and is therefore contained in some smallest closed interval, [nz(T), Af(T)], of RW. Also, ni(T) and Af(T) can he expressed in terms of the inner product on the Hilbert space. These along with some other properties of self-adjoint operators are stated in the following theorem Theorem 11. Let Te E (H) be closed interwel containing a(T). Then (i) nz(T) = inf (Th, h) (ii) Af (T) = sup (Th, h) and (iii) ||T|| = .sup |(Th, h)| = mesxr(|m(T)|, |AF(T)|) We define an operator, T, to be positive if (Th, h) > 0 for all h eH and we write T > 0. An operator T is positive if and only if T is self adjoint and a(T) c [0, 00) [47, p. :330]. It is easy to see that both TT* and T*T are positive operators for any Te E (H). If S, Te E (H) are two self-adjoint operators, we ;?-, S > T if and only if S T > 0. If we consider the scalar field to be RW instead of C, there may exist a positive operator T that is not self-adjoint. The following theorem can he found in [47] p. :331 [also see :37]. Theorem 12. Every positive operator T E B(H) has a unique positive square root S .such that S2 = T (184 U)C Urite S = Ti. If T is invertible. .so i~s S. Now, we describe the spectral theorem for normal operators which defines the operator f(T) where f is a bounded Borel measurable function defined on the spectrum, o-(T), of a bounded normal operator T. A nice exposition on spectral theory is given in Devito [9]. He starts with deriving the spectral theorem for operators on finite dimensional vector space in chapter 5 and ends in chapter 7 with spectral theorem for unbounded self-adjoint operator. There are various approaches to derive the spectral theorem for a bounded, normal operator. In this chapter we define spectral theorem following Rudin [47]. First, we define resolution of the .:ll. ,1.:1; Definition 5. Let B(R) be 8 : B(n) i (H) with the following properties: (i) (4) = 0. the zero operator. 8 (R) = I (ii) each 8(w) is (iii) S( (n w') = S (w) S(w') (iv) If w n w' = 4 then S( (U e') = S (w) + S (w') (v) for every g, h E H. the .set function. Seas. I' It...lI by is a complex measure on B(R). Since each S(w) is a self-adjoint projection, we have Simz( o) = (S(Lo)h, h) = ||(W~h||2 for all h E H. So from (v) it follows that Simit is a positive measure on B(R) for each hE H. Theorem 13. (Sp~ectral Theorem) Let T ES (H) be a normal operator. Then there exists a unique resolution of the .:1. ,:./:7;, 8, on B(a(T)) such that (Tg, h) =m AS,,(dA) for all g,h h H L/et f be a bounded Borel function on a(T). The operator f(T) is I. I;,.. l to be the unique operator ;oril r;.,ii l The above theorem justifies the following notations and f (T) = f I(A)Sl~dA) We refer to the above 8 as the spectral decomposition of T and we denote it by Sr. We now give an example. We know that any linear operator, T, on a finite dimensional vector space, V, can be represented as a matrix [9, p. 157]. We define T to be a diagonalizable operator if its matrix representation is a diagonalizable matrix. From [9, p. 167] we know that if T is a normal operator on V, then T is diagfonalizable. In particular, if we take V to be RW", then T can be represented by n x n diagonalizable matrices. Assume that all the eigenvalues of T are distinct and let Ai, A2, *, abe the distinct eigenvalues of T with corresponding eigenvectors P1, P2,..., P,. Hence, a(T) = {Az, a2, *, n} and B(a(T)) is the powe~rset, of e(T(.). For Ae B t(c(T)), we define Sr(Al) = i:AgEA Piri'. If A = weV define Sr(#) = 0, the zero operator. Since the eigenvectors Pi's are mutually orthogonal [9, p. 157], it is easy to see that Er(-) satisfies all the properties stated in Definition 5. Also note that, i= 1 1\ore generally, if T is a normal operator on a finite dimensional Hilbert space then Sr(A) is the orthogonal projection onto union of all eigenspaces for {Ag}, where As'sS are eigenvalues of T that are contained in ,4. The operator f(T) in Theorem 13 is of course a bounded operator and it can he proved that || f(T)|| < sup{| f ()| : Ae E (T)}. We also have the following theorem which determines the spectrum of the operator f(T). Theorem 14. Let Te E (H) be a normal operator and let f : a(T) C be continuous. Then f (T)* = f(T) || f(T)|| = mp{| f (A)| : Ae E (T)}. As a corollary of the above theorem we get the following useful result. Corollary 6. Let Te E (H) be a normal operator. Then |I ||= ||T ||' for every positive integer n. Now we introduce an important class of linear operators called compact operators. Recall that a subset ,4 of a Banach space is compact if every sequence {a,z} in 4 has a subsequence converging to some element of A. Definition 6. We n;, Te E (H) is compact ifT maps the unit ball. {h eH : ||h|| < 1} into a .set in H whose closure is compact. Retherford [37] gives a nice, concise description of compact operators. For a compact operator T, the spectrum, a(T), is at most countable, and has at most one limit point, namely, 0. Also, any non-zero value in the spectrum is necessarily an eigenvalue of T. We state these results in the following theorem ([8] p. 214, [9] chapter 5 ). Theorem 15. Let Te E (H) be a non-zero, compact, self adjoint operator. Let {A,} be the distinct, non-zero .:I c. ,:;rle;, of T arr rtg J:I~~ in such a way that I11 > I12 > i jfOri )j. and let {h,} be the corresponding sequence of orthonormal eigenvectors. If the sequence does not terminate, then lim |A,| = 0. Moreover, for h E H, ntOO n= 1 L/et G, be the eigenspace of As, i.e., G, is the closed linear span of all the As 's in {hm} having As as ~. .9. :;...le;, Then each G, has finite dimension and its dimension is the number of times An is repeated. F.:,..elle; the spectrum of T, o-(T), is ,.;.-l;; {A,} together with {0}. Some useful properties of compact operators are listed below: T is compact if and only if T* is compact. If T is compact and S E B(H) then TS and ST are compact operators. T is compact if and only if T*T is compact. T is compact if and only if TT* is compact. Every linear operator on a finite dimensional vector space is a compact operator and its spectrum coincides with the set of eigfenvalues of the operator. Another example of a compact operator is the Hilbert-Schmidt integral operator ([47] p. 112, [8] p. 267 ). Let (R, B(R), p) be a measure space. Suppose, t : O x R C be such that |tI (x, y) |2iilly< L Then the associated Hilbert-Schmidt integral operator is the operator T : L2( @ L2(2; 0,) giVen by (T f )(x) = tx )f()d~) 5.2 Application of Spectral Theory to Markov Chains Recall from C'!s Ilter 2 that P(:r, dy) is a Markov transition function on (X, B(X)) with invariant probability, measure, x Lt L (x) be the vector space of real-valued, measurable, mean-zero functions on X that are square-integrable with respect to xr, i.e., Ifw defin an;, inne produc~,,t on~2, L, (x y (,g) = x f(:r)g(:r)iT(dX) then L2, (x i Hilbert space with its norm given by || f|| = ( f, f) The Markov transition function, P(:r, dy) defines an operator, P, on L (xr). The operator P takes each f eL (xr) to the function, P~l fa\ (eL (x)), deindasfllows, By Jensen's inequality it follows that ||P|| < 1. So P is a bounded linear operator on L (x). The vector space L2- (x) is a subspa Ce of2(x) and' L (x) is the- spac tha~t is orthogonal to constant functions. We later describe the reason why we consider P to be an operator only on L2, (x)intead of the whole,, spac 2(). We would also like to mention that, unlike the previous section, here we consider the scalar field to be RW because in statistics we are mostly interested in real valued functions. In this case, the inner product is symmetric in its arguments i.e., ( f, g) = (g, f). Fr-om C'!s Ilter 2, we know that the Markov chain A (and so the transition function P(:r, dy)) is reversible if and only if for all f, y E L2(;~ (P f, g) =( f, Pg). It is easy to show that the above definition of reversibility is unchanged if the space L2(,) is replaced with L (xr). So, the Markov chain A is reversible if and only if P is a self-adjoint operator on L (xr). The operator I P, known as the Laplacian operator corresponding to P, pl .i-~ an important role in analyzing Markov chains and we denote this operator by lp. Note that the operator 1p is not one-to-one if there exists f (f 0) E L (x) such that Ip f= 0. But, from ?1. i-n and Tweedie's [1993] Proposition 17.4.1 it follows that for Harris ergodic Markov chains the only functions in L2 K) SatiSfying the equation Ip f = 0 are xr-almost everywhere constant. Since L (xr) does not include functions which are xr-almost everywhere constant (non-zero) and since we consider Harris ergodic Markov chains, the operator Ip is one-to-one in our case. But, Ip might not be an onto operator and so it might not be invertible. From the definition of spectrum of an operator we know that lp is invertible iff 1 / a(P). Since a(P) is a closed set it follows that 1 tf a(P) if and only if supa(P) < 1. Hence the Laplacian operator, lp, is invertible iff supa(P) < 1. Recall from C'!s Ilter 2 that we ;?- a CLT holds for fm (or simply f) if there exists a ,2 E (0, 00) such that, as m 00o, ~(~f Ex f) iN~(0, a"2 It is proved in [13] that a CLT holds for any function f lying in R(lp), the range of lp, and the corresponding .I-i-uspind ic~ variance in the CLT is given by v(f, P) = ||g||2_- l 12, where f = 1pg g EL (jT) So, if Ip is invertible, then CLT holds for every function feL1, (x. ls, from Teore ?,, we know that if ||P|| < 1, then Ip is alrv-ove invertible. Hence, we have the following proposition. Proposition 2. Let P be the M~arkov transition function of a Harris ergodic M~arkov chain with invariant I1:. e. -1 / If ||P || < 1, then CLT holds for,,,, evey untion fe (x .2, Remark 7. Note that, for a Harris ergodic M~arkov chain on a finite state space, lp is rl;, ate, invertible. In this case a(P) contains only the .:II no.;rle;,. of P and 1 tf a(P) because 1 is not an .:II ,.:.;rle;,. of P since we consider P as an operator on Li(xr). So, for a Harris ergodic M~arkov chain on a finite state space, CLT holds for every function fe U\L (x). Since Ip is one-to-one, even in the case when Ip is not onto we still can define the inverse operator ly] if the domain of ly D(lp ), ;?--, is restricted to R(lp) [32]. Fr-om [13] it then follows that CLT holds for all f e D(lfl) and by lemma 3.2 of Mira and G. Or-l [32] we have v(f P) = (f [21p' I]f ), V fe D (lf ). In C'!s Ilter 2, we defined the efficiency ordering due to [32]. The problem with efficiency ordering is that even when we conclude P FE Q, it might happen that v(f, P) = v(f, Q) for every f E L2(;T) that are of interest. In this chapter, we discuss conditions that guarantee v( f D,,. ~ P) < c v( f Q)f r evey eL(x (5-1) for some constant 0 < c < 1 and in this case we ;?-- P is more efficient than Q with uniform relative efficiency c. Also, we ;?i- that P is strictly more efficient than Q if vfC \, P) < v,,f,, Q)fr vry feL () (5-2) Note that (5-1) implies (5-2). But, (5-2) does not necessarily imply (5-1). Let lp and 1Q be the Laplacian operators corresponding to the Markov operators P and Q. If we assume that both lp and 1o are invertible then it follows that D(lp )= L~ (xr) D(lf ). Hence, we have v ( f, P) = ( f, [21-1 I] f), V fe ( x)- and v (j f ) = ( f [21,l I]) fo ), V fL (xr). Proposition 3. Assume that both lp and lq are invertible. Then v( f, P) < c v( f, Q) for every f E L2 K7) ;T C -1_ --1 + I ~~> 0 Proof. The proof follows from following equivalences v(f, P) m (f ,[2) -I]f ) < c(f, [21l If ) V fe L2~ + (f( c21,l I]- 21,1 I) f )>O Vfe L2r + c [21, I]-21,1 I> 0 + 2c ly1 21, + (1 c)I > 0 -1(1 c) W c 9 p + I '> 0. The above proposition is similar to Corollary 5.1 of Mira and G. Or-l [32]. Remark 8. In Proposition 8, if we replace the inequalities by strict inequalities and c by 1, we get conditions for strict e~ff. 7.: ,: ; For the rest of this section we assume that we have reversible Markov chains. We know that the Markov transition operator, P, of a reversible Markov chain is self-adjoint. So, we can use spectral theory to analyze reversible Markov chains. Let v( f, P) =,, 1 AS,7f~f(dA), (5-3) where 8(-) is the spectral decomposition of P. K~ipnis and Varadhan [24] show that if v( f, P) in (5-3) is finite, then there is a CLT for f with the .mon i nd il~lc variance, v( f, P) given by (5-3). Now onwards, we denote 87,7(-) by Syp(-). Recall from Chapter 2 that P is said to be better than Q in the efficiency ordering written P FE Q, if U f, P) I U f, Q) for every f E L2(x). It can be easily shown that the definition of efficiency ordering is unchanged if the space L2 Kr) ;,,1S, repace wth L (x). It is tempting to conclude, from (53) and th~e spectral thleoremn that P >E Q Iiff > ~. But, siinc~e o-(P) c [-1, 1], the function h(x) = is not necessarily bounded on o-(P) (o-(P) C [-1, 1] because ||P|| < 1). Hence we might not be able to use the spectral theorem to define h(P). The following definition is from Mira and G. Or-l [32]. Definition 7. Suppose, P and Q are two M~arkov transition functions with invariant I4~ll.:l..1l...;i measure xr. Then P is said to be better than Q2 in the covariance ordering written P >> Q, if Q > P on L (r) . It is easy to see that any Markov operator, P, dominates I, the identity operator, in Peskun ordering (Peskun [35], Tierney [50]). Also, Tierney [50] shows that Peskun ordering implies covariance ordering. So, P >> I, i.e., lp = I P is a positive operator. From our discussion in the previous section it then follows that there exists a unique square root 1) of Ip. K~ipnis and Varadhan [24] show that v(f, P) < 00 if and only if f E R(lf). Note that, this does not contradict the result of Gordin and Lif~sic [13] since D/l_) C R~lf). Mira, and/,, Geyrl [32]prov that P Fi Q if and only if P FE Q. We give an alternative proof for P >> Q implies P FE Q. The following theorem is from Bendat and Sherman [3] [see also 32, Theorem 4.1]. Theorem 16. Let h(x) be a bounded Borel measurable function on (lI, I2). A necessary and sufficient condition for h to have the 1/* */'' l ;, that h(S) < h(T) for all bounded, self-adjoint operators S, T with S < T and o-(S),o-(T) c (II, I2) iS tati h iS t'il: O (II, I2) Gnd Can it,'.ibil.:. .ill/ be continued into the whole upper-half plane, and represents there an r,:...;let..- function whose ie.:lrl.:..rry part is non-negative. Bendat and Sherman [3] mention that the function ax + b h(x) = with ad bc > 0 cx +d satisfies the condition of Theorem 16 either in x > -d or x < -. We now prove the following theorem. Theorem 17. Suppose that P and Q are two reversible M~arkov transition functions. If P Fi Qc then P FE -c) Proof. For E (0, 1) define 1 + x Comparing with the function h(:r) = @ ,w ae db 2( )an >1 o for e E (0, 1) the function h,(:r) is analytic on a(P) and a(Q) and it satisfies the conditions of Theorem 16. Since, we are assuming that P 1 IQ i.e., Q '> P, by Theorem 16 we have h,(Q) > h,(P) for ee (0, 1). For Ee (0, 1), h,(:r) is also a bounded Borel function on [-1,1]. So we can use the spectral theorem to define the operator h,(P). By spectral theorem, we have (h,(P) f, f) =~~ he(A)Syp~A). Let a+(P) = a(P) 0 [0, 1] and a-(P) = a(P) 0 [-1, 0) then we can write Note that, d 2A For Ae e*a(P), h,(A) > 0 and he(A) ? as e 1 Hence, by 1\lonotone convergence theorem, we have h.(A)Syp(dh) T1 1 /l + (P) + (P) For Ae t (P), he,(A) i \as e 1 and 0 < he,(A) < 1. Since S, 18yp(dAX) < || f ||2< 00 by Dominated convergence theorem we get, /,~ /,"EF~l~~ 1+Al- So, SJ,j h,(A)Syp(dAX) .Jb Sy~A asFI~, e 1. For e e (0, 1), we know that h,(Q) > h,(P) i.e. (ht(Q) f, f) >(h,(P) f, f) Vf fL (xr) 1.e., Then taking the limit e 1 on both sides we get i, 1+A_ P 1+A i \e., v Gf, Q) > vfP) Vf E O\L (x. O Proposition 4. Assume, sup?(a(P)) < 1 and sup?(a(Q)) < 1. Then P >> Q iff P FE -. Proof. It is assumed that sup(a(P)) < 1 and sup(a(Q)) < 1. So the function h.(x) = M is bounded on a(P) and a(Q). Since a(P), e(Q) C (-2, 1) and h(x) is analytic in x < 1, by Theorem 16 we have I +Qd I +P 2> Po > I I-Qd I- P Since a(P) < 1, the function h(x) is continuous on a(P). By Theorem 14, we know that h(P) is a bounded operator. Since we consider the scalar field to be RW, by Theorem 14, wve also know that h(P) is self-adljoint. Let g(x) = Then g(x) is analytic when x > -1. Let max(sup a(P), sup a(Q)) = e'. Then both a(h(P)) and a(h(Q)) are subsets of [0, ( )]. So, we can apply Theorem 16 on g to show I +Qd I +P > 4Q~2> P. I -Q d I- P Hence, we get I+Q I+P S> Po > ~ r -Q r -P The function h(x) = M is bounded on a(P) and a(Q). So, we can use spectral theorem to define the operator h.(P) = Then, byv (5 3) we have I+Q I+P > -- aP FE - I-Q r -P Hence , I +Qd I +P P ~ ~ ~ I-G)~ c ~>~P G) I d I - Now we discuss how theory of compact operators can he used to analyze Markov chains. Let P he the Markov transition function of a Harris ergodic Markov chain. Fr-on Proposition 2 we know that if ||P|| < 1, then CLT holds for every square integrable function f. It is easy to see that if P is a compact operator then ||P|| < 1. So, in order to establish CLT for a Markov chain we can show that the corresponding Markov transition operator, P, is compact. In Section 5.1, we mentioned that any Hilbert Schmidt operator is compact. Recall front ChI Ilter 2 that if a reversible Markov chain is geometrically ergodic, then the CLT holds for every square integrable function f [41]. Also we mentioned before that a reversible Markov chain, P, is geometrically ergodic if and only if ||P|| < 1 [41, 44]. Schervish and Carlin [48] and Liu et al. [26]) have proved geometric ergodicity of certain Markov chains by establishing that the corresponding Markov operator, P, is Hilbert Schmidt. APPENDIX: CHEN AND SHAO'S CONDITIONS Here we state C'I. n! and Shao's [2000] necessary and sufficient conditions for city) < 00 as well as a simple method for checking these conditions. Let X denote the nx p matrix whose ith row is XT and let W denote an n x p matrix whose ith row is WT, where Xi if yi = 0 S-Xi if yi = 1. Proposition 5. [6] The function city) is finite if and only if (i) the design matrix: X has full column rank, and (ii) there exists a vector a = (al,..., a,) with strictly positive components such that W~a = 0 . 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[:32] Mira, A. and Geyer, C. J. (1999), "Ordering Monte Carlo Markov chains," Tech. Rep.
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[:33] Mykland, P., Tierney, L., and Yu, B. (1995), "Regeneration in Markov chain
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[:34] Nummelin, E. (1984) General Irreducible Iabrkov C'hain~s and Non-negative Op~era-
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[:39] Robert, C. P. (1995), "Simulation of truncated normal variables," Stratistic~s and

[40] Roberts, G. and Tweedie, R. (1999), "Bounds on regeneration times and convergence
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[41] Roberts, G. O. and Rosenthal, J. S. (1997), "Geometric ergodicity and hybrid Markov
chains," Electronic C'otmunications in Pr o~l~.:l..7.;; 2, 1:325.

[42] -(2001), :\1 I) 1:0v C'I !.1.4 and de-initializing processes," Scandinavian Journal of
Shetistic~s, 28, 489-504.

[4:3] -(2004), "General State Space Markov ('1. !.1.4 and MC' lC Algorithms," Pr o~l~.,t.7. /
Surveys, 1, 20-71.

[44] Roberts, G. O. and Tweedie, R. L. (2001), "Geometric L2 and L1 convergence are
equivalent for reversible Markov chains," Jourmal of Applied P l~r l~.:l..7.;; :38A, :37-41.

[45] Rosenthal, J. S. (1995), il!s...il. II!. .1, Conditions and Convergence Rates for Markov
C'I I!1, Monte Carlo," Journal of the American Statistical A~ssociation, 90, 558-566.

[46] Roy, V. and Hohert, J. P. (2007), "Convergence rates and .I-i nspinile; 1 standard errors
for MC1|LC algorithms for B li- -i Ils profit regression," Journal of the Roriarl Statistical
S .. .:. It; Series B, 69, 607-623.

[47] Rudin, W. (1991), Functional A,:tale;,.: McGraw-Hill, 2nd ed.

[48] Schervish, 31. J. and Carlin, B. P. (1992), "On the Convergence of Successive
Substitution CI .visplilrs Joureml of C'omputational and Grap~hical Stratistic~s, 1,
111-127.

[49] Tierney, L. (1994), 11 I) In~v chains for exploring posterior distributions (with
discussion)," The Annedls of Shetistic~s, 22, 1701-1762.

[50] -(1998), "A Note on Metropolis-Hastings K~ernels for General State Spaces," The
Annals of Applied Pr o~l~.:l..7.;; 8, 1-9.

[51] van Dyk, D. A. and Meng, X.-L. (2001), "The Art of Data Augmentation (with
Discussion) ," Jourmal of C'omputatiomel and Grap~hical Stratistic~s, 10, 1-50.

BIOGRAPHICAL SKETCH

Mr. Vivekananda Roy was born in 1980, in West Bengal, India. He spent his

childhood in his ancestral village, Bachhipur, where he went to school. After passing the

higher secondary examination, he moved to Calcutta in 1998. He received his bachelor's

degree in statistics from the University of Calcutta in 2001. He then joined the Indian

Statistical Institute, from where he received his Master of Statistics degree in 2003. He

joined the graduate program of the Department of Statistics at University of Florida in fall

2003. Upon graduation from UF, he will join the Department of Statistics at lowa State

University, as an Assistant Professor.

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 ABSTRACT ........................................ 7 CHAPTER 1INTRODUCTION .................................. 9 2MARKOVCHAINBACKGROUND ........................ 18 3BAYESIANPROBITREGRESSION ........................ 24 3.1Introduction ................................... 24 3.2GeometricConvergenceandCLTsfortheACAlgorithm .......... 27 3.3ComparingtheACandPX-DAAlgorithms ................. 35 3.4ConsistentEstimatorsofAsymptoticVariancesviaRegeneration ...... 37 4BAYESIANMULTIVARIATEREGRESSION 48 4.1Introduction ................................... 48 4.2ProofofPosteriorPropriety .......................... 50 4.3TheAlgorithms ................................. 58 4.3.1DataAugmentation ........................... 58 4.3.2HaarPX-DAAlgorithm ......................... 60 4.4GeometricErgodicityoftheAlgorithms .................... 61 5SPECTRALTHEOREMANDORDERINGOFMARKOVCHAINS ...... 71 5.1SpectralTheoryforNormalOperators .................... 71 5.2ApplicationofSpectralTheorytoMarkovChains .............. 81 APPENDIX:CHENANDSHAO'SCONDITIONS ................... 89 REFERENCES ....................................... 90 BIOGRAPHICALSKETCH ................................ 94 5

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Table page 3-1ResultsbasedonR=100regenerations ...................... 47 6

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2exp 10

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2exp 2,theposteriordensitytakesthefollowingform(;jy)=1 2exp 2wherec2(y)isthemarginaldensityofygivenbyc2(y)=ZWZRdknYi=1Z10d 2exp 2dd;whereWRd(d+1) 2isthesetofddpositivedenitematrices.Inchapter4,weprovidenecessaryandsucientconditionsforc2(y)<1.Asinthepreviousexample,posteriorexpectationswithrespecttotheposteriordensity,(;jy),arenotavailableinclosed-form.Wenowdiscussdierentcomputationalmethodsthatcanbeusedtoapproximate( 1{1 ).Thesecomputationalmethodsarebroadlyoftwotypes,namely,numericalintegrationmethodsandsimulationbasedmethods.Ifthedimension,p,isnotlarge, 11

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1{1 )bysimulatingaMarkovchainwithstationarydistribution.ThisisthebasicprincipleofMarkovchainMonteCarlo(MCMC)method.ThemostgeneralalgorithmforproducingMarkovchainswitharbitrarystationarydistributionistheMetropolis-Hastings(M-H)algorithm.AsimpleintroductiontotheM-HalgorithmisgiveninChibandGreenberg[ 7 ].Another 13

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4 ].Supposethep-dimensionalvectorin( 1{1 )canbewrittenas=(1;2;:::;p).ThesimplestGibbssampler(but,notthegeneralGibbssampler)requiresonetobeabletosimulatefromallunivariatefullconditionaldensitiesofi.e.,itisrequiredtosimulatefromtheconditionaldistributions,ijfj;j6=igfori=1;2;:::;p.ItisalsopossibletocreateahybridalgorithmwhichusesdierentversionsofM-HalgorithmtogetherwithGibbssamplertoconstructaMarkovchainwithstationarydistribution.Asourdiscussionsuggests,thereisaplethoraofMarkovchainswithstationarydistribution.InordertochoosebetweenMCMCalgorithms,weneedanorderingofMarkovchainshavingthesamestationarydistribution.InChapters2and5,wedescribedierentpartialorderingsofMarkovchains.LetfXjg1j=0denotetheMarkovchainassociatedwithanMCMCalgorithmthatisusedtoexplore.IffXjg1j=0isHarrisergodic(denedinChapter2),theergodictheoremimpliesthat,nomatterwhatthedistributionofthestartingvalue,X0, 14

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AlbertandChib 's[ 1993 ]dataaugmentationalgorithmand LiuandWu 's[ 1999 ]PX-DAalgorithm.Westudytheconvergencerateofthesealgorithmsandprovetheexistenceofcentrallimittheorems(CLTs)forergodicaveragesunderasecondmomentcondition.WecomparethesetwoalgorithmsandshowthatthePX-DAalgorithmshouldalwaysbeusedsinceitismoreecientthantheotheralgorithminthesenseofhavingsmallerasymptoticvarianceinthecentrallimittheorem(CLT).Asimple,consistentestimatoroftheasymptoticvarianceintheCLTisconstructedusingregenerativesimulationmethods.InChapter4,weconsiderBayesianmultivariateregressionmodelswherethedistributionoftheerrorsisascalemixtureofnormals.Wenoticedbeforethatifthestandardnoninformativepriorisusedontheparameters(;),thenposteriorexpectationswithrespecttothecorrespondingposteriordensity,(;jy),arenotavailableinclosed-form.WedeveloptwoMCMCalgorithmsthatcanbeusedtoexplorethedensity(;jy).ThesealgorithmsarethedataaugmentationalgorithmandtheHaarPX-DAalgorithm.Wecomparethetwoalgorithmsandstudytheirconvergerates.Wealsoprovidenecessaryandsucientconditionsfortheproprietyoftheposteriordensity,(;jy).WhileinChapters3and4,weusedprobabilistictechniquestoanalyzedierentMCMCalgorithms,itispossibletotakeafunctionalanalyticapproachtostudyandcomparedierentMarkovchains.InChapter5,wegiveabriefoverviewofsomeresultsfromfunctionalanalysis.Inparticular,wediscussthespectraltheoremforbounded, 16

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Rosenthal 's[ 1995 ]Theorem12showsthattheabovedriftandminorizationconditions,together,implythatisgeometricallyergodic.InChapter4,weemploydriftandminorizationconditionstoprovethegeometricergodicityofthedataaugmentationalgorithmusedinBayesianmultivariateStudent'stregressionproblem.Oneadvantageofprovinggeometricergodicityofbyestablishingtheabovedriftandminorizationconditionsisthatusing Rosenthal 's[ 1995 ]Theorem12,wealsocancalculateanupperboundofM(x)min( 2{1 ).Thisupperboundcanbeusedtocomputeanappropriateburn-inperiod(JonesandHobert[ 23 ],MarchevandHobert[ 28 ]).ThereareothermethodsofprovinggeometricergodicityofaMarkovchainthatdonotprovideanyquantitativeboundofM(x)min( 2{1 ).Wedescribeonesuchmethodnow.WewillassumethatXisequippedwithalocallycompact,separable,metrizabletopologywithB(X)astheBorel-eld.AfunctionV:X![0;1)issaidtobeunboundedocompactsetsifforevery>0,thelevelsetfx:V(x)giscompact.TheMarkovchainissaidtobeaFellerchainif,foranyopensetO2B(X),P(;O)isalower-semicontinuousfunction.Thefollowingpropositionisaspecialcaseof MeynandTweedie 's[ 1993 ]Lemma15.2.8.

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1 toestablishgeometricergodicityofMCMCalgorithmsusedinBayesianprobitregressionproblem.HobertandGeyer[ 15 ]employedProposition 1 toestablishthegeometricergodicityofGibbssamplersassociatedwithBayesianhierarchicalrandomeectsmodels.Noticethat,unlikeProposition 1 ,thedriftconditionin Rosenthal 's[ 1995 ]Theorem12doesnotrequirethedriftfunction,V,tobeunboundedocompactsets.Also, Rosenthal 's[ 1995 ]Theorem12doesnotneedtobeaFellerchain.ThedrivingforcebehindMCMCistheergodictheorem,whichissimplyaversionofthestronglawthatholdsforwell-behavedMarkovchains;e.g.,HarrisergodicMarkovchains.Indeed,supposethatf:X!RissuchthatRXjfjd<1anddeneEf=RXfd.Thentheergodictheoremsaysthattheaverage 41 ].(FormoreontheCLTinMCMC,seeChanandGeyer[ 5 ],MiraandGeyer[ 32 ],Jones[ 20 ]andJonesetal.[ 21 ].)ForathoroughdevelopmentofgeneralstatespaceMarkovchaintheory,seeNummelin[ 34 ]andMeynandTweedie[ 30 ].RobertsandRosenthal[ 43 ]providesaconcise,self-containeddescriptionongeneralstatespaceMarkovchains(alsoseeTierney[ 49 ]).AsmentionedinChapter1,foragivendistributionfunction,,therearelargenumberofMCMCalgorithmswithstationarydistribution.Onewaytoorderthese 22

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6 ]providenecessaryandsucientconditionsonyandfxigni=1forc1(y)<1andtheseconditionsarestatedexplicitlyintheAppendix.Whentheseconditionshold,theposteriordensityofiswelldened(i.e.,proper)andisgivenby(jy)=1 AlbertandChib 's[ 1993 ]dataaugmentationalgorithm,whichwenowdescribe.LetXdenotethenpdesignmatrixwhoseithrowisxTiand,forz=(z1;:::;zn)T2Rn,let^=^(z)=(XTX)1XTz.Also,letTN(;2;w)denoteanormaldistributionwith 24

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(i) Drawz1;:::;znindependentlywithziTN(xTi;1;yi) (ii) Draw0Np^(z);(XTX)1AlbertandChib[ 1 ]hasbeenreferencedover350times,whichshowsthattheACalgorithmanditsvariantshavebeenwidelyappliedandstudied.ThePX-DAalgorithmofLiuandWu[ 27 ]isamodiedversionoftheACalgorithmthatalsosimulatesaMarkovchainwhoseinvariantdensityis(jy).AsingleiterationofthePX-DAalgorithmentailsthefollowingthreesteps: (i) Drawz1;:::;znindependentlywithziTN(xTi;1;yi) (ii) Drawg2Gamman 2Pni=1zixTi(XTX)1XTz2andsetz0=(gz1;:::;gzn)T Draw0Np^(z0);(XTX)1NotethattherstandthirdstepsofthePX-DAalgorithmarethesameasthetwostepsoftheACalgorithmso,nomatterwhatthedimensionof,thedierencebetweentheACandPX-DAalgorithmsisjustasingledrawfromtheunivariategammadistribution.Fortypicalvaluesofnandp,theeortrequiredtomakethisextraunivariatedrawisinsignicantrelativetothetotalamountofcomputationneededtoperformoneiterationoftheACalgorithm.Thus,thetwoalgorithmsarebasicallyequivalentfromacomputationalstandpoint.However,LiuandWu[ 27 ]andvanDykandMeng[ 51 ]bothprovideconsiderableempiricalevidencethatautocorrelationsdiedownmuchfasterunderPX-DAthanunderAC,whichsuggeststhatthePX-DAalgorithm\mixesfaster"thantheACalgorithm.(LiuandWu[ 27 ]alsoestablishedatheoreticalresultalongtheselines-seetheproofofourCorollary 1 .) 25

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3{1 )isbyestablishinggeometricergodicityoffjg1j=0.Inthischapter,weprovethattheMarkovchainsunderlyingtheACandPX-DAalgorithmsbothconvergeatageometricratewhichimpliesthattheCLTin( 3{1 )holdsforeveryf2L2(jy);thatis,foreveryfsuchthatRRpf2()(jy)d<1.WealsoestablishthatPX-DAistheoreticallymoreecientthanACinthesensethattheasymptoticvarianceintheCLTunderthePX-DAalgorithmisnolargerthanthatundertheACalgorithm.Regenerativemethodsareusedtoconstructasimple,consistentestimatoroftheasymptoticvarianceintheCLT.Asanillustration,weapplyourresultsto vanDykandMeng 's[ 2001 ]lupusdata.Inthisparticularexample,theestimatedasymptoticrelativeeciencyofthePX-DAalgorithmwithrespecttotheACalgorithmisabout65.Hence,eventhoughtheACandPX-DAalgorithmsareessentiallyequivalentintermsofcomputationalcomplexity,hugegainsineciencyarepossiblebyusingPX-DA. 26

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3.2 and 3.3 ,respectively.InSection 3.4 wederiveresultsthatallowfortheconsistentestimationofasymptoticvariancesviaregenerativesimulation.

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34 ,Theorem3.8].Supposehisabounded,harmonicfunction.SincetheACalgorithmis-irreducibleandhasaninvariantprobability 28

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34 ,Proposition3.13].Thus,thereexistsasetNwith(N)=0suchthath()=cforall2 2 itfollowsthatergodictheoremholdsforit.Thefollowingtheoremisthemainresultofthissection. Proof. 1 .WehaveshownthatACalgorithmis-irreducibleandaperiodic,wheredenotetheLebesguemeasureonRp.So,ifisamaximalirreducibilitymeasurefortheMarkovchainunderlyingtheACalgorithm,then.Conversely,if(A)=0,thenKm(;A)=0forall2Rpandallm2N,whichimpliesthat(A)=0anditfollowsthat.Hence,.Sincethesupportofobviouslyhasnon-emptyinterior,itfollowsthatthesupportofamaximalirreducibilitymeasurefortheACalgorithmhasnon-emptyinterior.WenowdemonstratethattheMarkovchainassociatedwiththeACalgorithmisaFellerchain.LetandOdenoteapointandanopensetinRp,respectively.Assumethatfmg1m=1;isa(deterministic)sequenceinRpwithm6=suchthatm!asm!1.TwoapplicationsofFatou'sLemmainconjunctionwiththefactthat(zj;y)iscontinuousinyieldliminfm!1K(m;O)ZOliminfm!1k(0jm)d0=ZOliminfm!1"ZRm(0jz;y)(zjm;y)dz#d0

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1 withdriftfunctionV()=(X)T(X).RecallthatXisassumedtohavefullcolumnrank,p,andhenceXTXispositivedenite.Thus,foreach>0,thesetf2Rp:V()g=f2Rp:T(XTX)giscompactsothefunctionVisunboundedocompactsets.Now,notethat(KV)()=ZRpV(0)k(0j)d0=ZRp"ZRnV(0)(0jz;y)d0#(zj;y)dz=ZRpEV(0)jz;y(zj;y)dz=EnEV(0)z;y;yo;where,asthenotationsuggests,theexpectationsinthelasttwolinesarewithrespecttotheconditionaldensities(0jz;y)and(zj;y).Recallthat(0jz;y)isap-dimensionalnormaldensityand(zj;y)isaproductoftruncatednormals.Evaluatingtheinsideexpectation,wehaveEV(0)z;y=E(0)TXTX0z;y=tr(XTX(XTX)1)+zTX(XTX)1(XTX)(XTX)1XTz=p+zTX(XTX)1XTzp+zTz;

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19 ]implythatifUTN(;1;1)then,E(U2)=1+2+() ();where()withonlyasingleargumentdenotesthestandardnormaldensityfunction;thatis,(v)isequivalentto(v;0;1).Similarly,ifUTN(;1;0)then,E(U2)=1+2() 1():ItfollowsthatEz2ij;y=8><>:1+(xTi)2+(xTi)(xTi) (xTi)ifyi=11+(xTi)2(xTi)(xTi) 1(xTi)ifyi=0:Amorecompactwayofexpressingthisisasfollows: Ez2ij;y=1+(wTi)2(wTi)(wTi) 1(wTi);(3{2)wherewiisdenedintheAppendix.Hence,wehave (KV)()=EnEV(0)z;y;yop+n+nXi=1(wTi)2nXi=1(wTi)(wTi) 1(wTi):(3{3)Recallthatthegoalistoshowthat(KV)()V()+Lforall2Rp.Itfollowsfrom( 3{3 )that(KV)(0)p+n.Wenowconcentrateon2Rpnf0g.WebeginbyconstructingapartitionofthesetRpnf0gusingthenhyperplanesdenedbywTi=0.Forapositiveintegerm,deneNm=f1;2;:::;mg.LetA1;A2;:::;A2ndenoteallthesubsetsofNn,and,foreachj2N2n,deneacorresponding 31

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[2nj=1Sj=Rpnf0g,and 5 areinforce,thereexiststrictlypositiveconstantsfaigni=1suchthata1wT1+a2wT2++anwTn=0:Therefore, 3{4 )impliesthattheremustalsoexistani06=isuchthatwTi0andwTihaveoppositesigns.Thus,AjandAjarebothnonempty.NowdeneC=j2N2n:Sj6=;.Foreachj2C,deneRj()=Pi2Aj(wTi)2 11 ,p.175].Also,itisclearthatif 32

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1(u);thenM2(0;1).Fixj2C.Itfollowsfrom( 3{3 )andtheresultsconcerningMillsratiothatforall2Sj,wehave(KV)()p+n+nXi=1(wTi)2Xi2Aj(wTi)(wTi) 1(wTi)Xi2 1(wTi)p+n+nXi=1(wTi)2+Xi2Aj(wTi)(wTi) 1(wTi)Xi2 1(wTi)p+n+nXi=1(wTi)2+nMXi2

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10 ]forbackgroundonleftgroupactionsandmultipliers.)LetZdenotethesubsetofRninwhichzlives;i.e.,ZistheCartesianproductofnhalf-lines(R+andR),wheretheithcomponentisR+ifyi=1andRifyi=0.Fixz2Z.ItiseasytoseethatStep2ofthePX-DAalgorithmisequivalenttothetransitionz!gzwheregisdrawnfromadistributiononGhavingdensityfunction 17 ].First,theirProposition3showsthatR(z;dz0)isreversiblewithrespectto(zjy)anditfollowsthatk(0j)isreversiblewithrespectto(jy).WenowusethefactthattheACalgorithmisgeometricallyergodictoestablishthatthePX-DAalgorithmenjoysthispropertyaswell. Proof. 25 32 ].DenotethenormsoftheseoperatorsbykKkandkKk.Ingeneral,areversible,HarrisergodicMarkovchainisgeometricallyergodicifandonlyifthenormoftheassociatedMarkovoperator 36

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41 44 ].ByTheorem 1 ,theACalgorithmisgeometricallyergodicandconsequentlykKk<1.ButLiuandWu[ 27 ]showthatkKkkKk[seealso 17 ,Theorem4]andhencekKk<1,whichimpliesthatthePX-DAalgorithmisalsogeometricallyergodic. WehavenowshownthattheMarkovchainsunderlyingtheACandPX-DAalgorithmsarebothreversibleandgeometricallyergodicandhencebothhaveCLTsforallf2L2(jy).WenowuseanotherresultfromHobertandMarchev[ 17 ]toshowthatthePX-DAalgorithmisatleastasecientastheACalgorithm. Proof. HobertandMarchev 's[ 2008 ]Theorem4. Inordertoputourtheoreticalresultstouseinpracticetocomputevalidasymptoticstandarderrors,werequireaconsistentestimatoroftheasymptoticvarianceandthisisthesubjectofthenextsection. 3 .Ofcourse,themarginalchainfjg1j=0hastheMarkovtransitiondensityk(0j)nomatterwhichversionoftheMarkovtransitiondensitywechooseforthejointchain.Whilethisisobviousforthechaincorresponding~~k,itcanbeeasilyshownfor~kbyconsideringtwoconsecutivesteps 37

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42 ]canbeusedtoshowthatthejointchainfj;zjg1j=0inheritsgeometricergodicityfromitsmarginalchainfjg1j=0.Notethatfj;zjg1j=0isthechainthatisactuallysimulatedwhentheACalgorithmisrun(wejustignorethezjs).Supposewecanndafunctions:RpRn![0;1],whoseexpectationwithrespectto(;zjy)isstrictlypositive,andaprobabilitydensityd(0;z0)onRpRnsuchthatforall(0;z0);(;z)2RpRn,wehave ~k0;z0j;zs(;z)d(0;z0):(3{6)Thisiscalledaminorizationcondition[ 22 30 43 ]anditcanbeusedtointroduceregenerationsintotheMarkovchaindrivenby~k.Theseregenerationsarethekeytoconstructingasimple,consistentestimatorofthevarianceintheCLT.Afterexplainingexactlyhowthisisdone,wewillidentifysanddforbothACandPX-DA.Equation( 3{6 )allowsustorewrite~kasthefollowingtwo-componentmixturedensity ~k0;z0j;z=s(;z)d(0;z0)+1s(;z)r0;z0j;z;(3{7) 38

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1s(;z);whens(;z)<1(anddenedarbitrarilywhens(;z)=1).InsteadofsimulatingtheMarkovchainfj;zjg1j=0intheusualwaythatalternatesbetweendrawsfrom(zj;y)and(jz;y),wecouldsimulatethechainusingthemixturerepresentation( 3{7 )asfollows.Supposethecurrentstateis(j;zj)=(;z).First,wedrawjBernoullis(;z).Thenifj=1,wedraw(j+1;zj+1)fromd,andifj=0,wedraw(j+1;zj+1)fromtheresidualdensity.The(random)timesatwhichj=1correspondtoregenerationsinthesensethattheprocessprobabilisticallyrestartsitselfatthenextiteration.Morespecically,supposewestartbydrawing(0;z0)d.Theneverytimej=1,wehave(j+1;zj+1)dsotheprocessis,ineect,startingoveragain.Furthermore,the\tours"takenbythechaininbetweentheseembeddedregenerationtimesareiid,whichmeansthatstandardiidtheorycanbeusedtoanalyzetheasymptoticbehaviorofergodicaverages,therebycircumventingthedicultiesassociatedwithanalyzingaveragesofdependentrandomvariables.Formoredetailsandsimpleexamples,seeMyklandetal.[ 33 ]andHobertetal.[ 16 ].Inpractice,wecanevenavoidhavingtodrawfromr(whichcanbeproblematic)simplybydoingthingsinaslightlydierentorder.Indeed,giventhecurrentstate(j;zj)=(;z),wedraw(j+1;zj+1)intheusualway(thatis,bydrawingfrom(zj;y)and(jz;y))afterwhichwe\llin"avalueforjbydrawingfromtheconditionaldistributionofjgiven(j;zj)and(j+1;zj+1),whichisjustaBernoullidistributionwithsuccessprobabilitygivenby ~k(j+1;zj+1jj;zj):(3{8) 39

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N=1 16 ]areapplicableandimplythat,aslongasthereexistsan>0suchthatEjf()j2+jy<1,then N2: 3{9 )isslightlydierentfromtheCLTdiscussedearlier,whichtakestheformp

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(2)p 2exp1 2^(z)TXTX^(z):where^(z)=(XTX)1XTzThus,s(z)="inf2D(jz;y) 2^(z)TXTX^(z)2TXTX^(z) 2^(z)TXTX^(z)2TXTX^(z)="exp1 2zTX(XTX)1XTz 2zTX(XTX)1XTzinf2Dexp(zz)TX="exp1 2zTX(XTX)1XTz 2zTX(XTX)1XTzexppXi=1citiIR+(ti)+ditiIR(ti)

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~k(j+1;(zj+1;gj+1)jj;(zj;gj))=inf2D(jgjzj;y) 2g2jzTjX(XTX)1XTzj 2zTX(XTX)1XTzexppXi=1cit(j)iIR+(t(j)i)+dit(j)iIR(t(j)i)exp1 2zTX(XTX)1XTz 2g2jzTjX(XTX)1XTzjexp(gjzjz)TXj+1ID(j+1)=exp(pXi=1cit(j)iIR+(t(j)i)+dit(j)iIR(t(j)i)t(j)ij+1;i)ID(j+1)wheret(j)T=(gjzjz)TX.Theorem 2 statesthattheasymptoticvarianceintheCLTforthePX-DAalgorithmisnolargerthanthatfortheACalgorithm;i.e,0<2f;k2f;k<1forallf2L2(jy).However,weknowfromRemark 1 thattheregenerativemethodisbasedonaslightlydierentCLTwhoseasymptoticvariancehasanextrafactorinvolvingthesmallfunction,namelyE(s()jy),fromtheminorizationcondition.AlthoughthesmallfunctionsinthetwominorizationconditionsthatwederivedfortheACandPX-DAalgorithmsareslightlydierent,E(s(:)jy)remainsexactlythesameasshown 45

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2(n2)=2(n=2)ZRnZR+z0T(IH)z0n vanDykandMeng 's[ 2001 ]lupusdata,whichconsistsoftriples(yi;xi1;xi2),i=1;:::;55,wherexi1andxi2arecovariatesindicatingthelevelsofcertainantibodiesintheithindividualandyiisanindicatorforlatentmembranouslupusnephritis(1forpresenceand0forabsence).vanDykandMeng[ 51 ]consideredthemodelPr(Yi=1)=0+1xi1+2xi2;withaatprioron.Weusedalinearprogram(thatisdescribedintheAppendix)toverifythat ChenandShao 's[ 2000 ]necessaryandsucientconditionsforproprietyaresatisedinthiscase. 46

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39 ]togenerateone-sidedtruncatednormalrandomvariables.WeranACandPX-DAforR=100regenerationseach.Thistook1,223,576iterationsforACand1,256,677iterationsforPX-DA.WeusedthesimulationstoestimatetheposteriorexpectationsoftheregressionparametersandtheresultsareshowninTable 3-1 .(ResultsinChenandShao[ 6 ]implythatthereexists>0suchthatEjjj2+jy<1forj2f0;1;2g.)ItisstrikingthattheestimatedasymptoticvariancesfortheACalgorithmareallatleast65timesaslargeasthecorrespondingvaluesforthePX-DAalgorithm.Theseestimatessuggestthat,inthisparticularexample,theACalgorithmrequiresabout65timesasmanyiterationsasthePX-DAalgorithmtoachievethesamelevelofprecision.(Weactuallyrepeatedtheentireexperimentseventimesandtheestimatesof2f;k=2f;krangedbetween40and145.) Table3-1. ResultsbasedonR=100regenerations ACAlgorithmPX-DAAlgorithm Parameter estimates.e.^f;k=p -3.0180.012 66.61 6.9160.023 66.92 3.9820.015 63.1 47

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2exp 4{1 )asy=X+"wherey=(y1;:::;yn)Tisthendmatrixofobservations,X=(x1;x2;:::;xn)Tisthenkmatrixofcovariatesand"=("1;:::;"n)Tisthendmatrixoferrorvariables.Thelikelihoodfunctionfortheregressionmodelin( 4{1 )isgivenbyf(yj;)=nYi=1Z10d 2exp

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2.Theposteriordensitytakesthefollowingform 2isthesetofddpositivedenitematrices.FernandezandSteel[ 12 ]provedthatc2(y)<1ifandonlyifnd+k.Insection 4.2 ,wegiveanalternativeproofoftheposteriorpropriety.Abyproductofourproofisamethodofexactsamplingfrom(;jy)intheparticularcasewhennisexactlyd+k.Throughoutthischapterweassumethatnd+k.Wealsoassumethatthecovariatematrix,X,isoffullcolumnranki.e.,r(X)=k.Theposteriordensityin( 4{3 )isintractableinthesensethatposteriorexpectationsarenotavailableinclosed-form.Also,ourexperienceshowsthatitisdiculttodevelopausefulprocedureformakingi.i.d.drawsfrom(;jy).InthischapterwefocusonMCMCmethodsforexploringtheposteriordensityin( 4{3 ).Wedevelopadataaugmentation(DA)algorithmfor(;jy)insection 4.3.1 .IthasbeennoticedintheliteraturethatthestandardDAalgorithmoftensuersfromslowconvergence[ 51 ].EmpiricalandtheoreticalstudieshaveshownthatalternativealgorithmsthataremodiedversionsofthestandardDAalgorithmsuchastheHaarPX-DAalgorithmandthemarginalaugmentationalgorithmoftenprovidehugeimprovementoverthestandardDAalgorithm(LiuandWu[ 27 ],vanDykandMeng[ 51 ],RoyandHobert[ 46 ],HobertandMarchev[ 17 ]).Insection 4.3.2 ,wedeveloptheHaarPX-DAalgorithmfortheposteriordensityin( 4{3 ).Wethenspecializetothecasewhentheerrors,"i's,haveaStudent'stdistributioni.e.,themixingdistribution,H(),in( 4{2 )isaGamma( 49

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17 ]wealsoconcludethattheHaarPX-DAalgorithmisatleastasecientasthedataaugmentationalgorithminthesensethattheasymptoticvariancesinthecentraltheoremunderHaarPX-DAalgorithmareneverlargerthanthoseundertheDAalgorithm.SomeoftheseresultsaregeneralizationsofresultsfromvanDykandMeng[ 51 ]andMarchevandHobert[ 28 ]whoconsideredthespecialcasewheretherearenocovariatesintheregressionmodel( 4{1 ),i.e.,X=(1;1;:::;1)TandHisGamma( 4.3 ,wedescribetheDAandtheHaarPX-DAalgorithms.Inthelastsection,wecomparethetwoalgorithmsandprovethatboththealgorithmsconvergeatageometricrate. 50

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4{1 )and( 4{2 )itfollowsthat (2)nd 2:

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2 ,Chapter17] 2 ]Supposeisanmrmatrix,andaremmandrrnon-negativedenitematrices.WesaythatZhasamatrixnormaldistributionwithparameters,andifZisanmrrandommatrixhavingmomentgeneratingfunctiongivenbyMZ(t)=exptr(Tt)+1 2tr(tTt)andwewriteZNm;r(;;).InthiscasewehaveE(Z)=.Moreover,ifandarepositivedenitematricesthenZhasthefollowingdensityfunctionfZ(z)=1 (2)mr=2jjr=2jjm=2exp1 2tr1(z)1(z)T:Sincer(X)=k,itfollowsthatXTQ1Xandhenceisap.d.matrix.Thus, 2tr1yTQ1yT1)jQjd 2

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2 2exp(1 2tr1yTQ1yT1)jjd 2IQ1 2X(XTQ1X)1XTQ1 2Q1 2y andsinceIQ1 2X(XTQ1X)1XTQ1 2isanidempotentmatrix,itimpliesthatyTQ1yT1isapositivesemi-denitematrix.NowweprovethatyTQ1yT1isap.d.matrixbyshowingthatjyTQ1yT1j6=0(withprobabilityone).Letbethen(d+k)augmentedmatrix(X:y).Then,TQ1=264XTyT375Q1Xy=264XTQ1XXTQ1yyTQ1XyTQ1y375:Therefore, Sincer(X)=k,weknowthatXTQ1Xisap.d.matrixandhencejj>0.Alsosincend+k,thed+kcolumnsofarelinearlyindependentwithprobabilityonebecausetheprobabilityofndimensionalrandomcolumnvectorsofylyinginanylinearsubspaceofRnwithdimensionn1iszero(withrespecttotheLebesguemeasureonRdn). 53

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4{8 )itfollowsthatjyTQ1yT1j>0.Tointegratetheexpressionin( 4{6 )withrespectto,weusethefollowingdenitionofInverseWishartdistribution. 29 ,p.85]Letbeappp.d.matrix.ThenforsomempthepprandommatrixWissaidtohaveaInverseWishartdistributionwithparametersmandifthep.d.fofW(withrespecttoLebesguemeasureonRp(p+1) 2,restrictedtothesetwhereW>0)isgivenbyf(W;m;)=jWjm+p+1 2exptr(1W1) 2 4jjm 2(m+1i))andwewriteWIWp(m;).Henceifnd+ki.e.,nkdbytheabovedenitionofInverseWishartdistribution,wehave =2d(nk) 2d(d1) 4Qdi=1(1 2(nk+1i)) (2)d(nk) 2yTQ1yT1nk 2(nk+1i)) 4yTQ1yT1nk 4{8 )weget 4jTQ1jd 4jjdnYj=1h(qj): 54

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4jjd;which,ofcourse,isanitenumber.So,wehaveprovedthatintheparticularcasewhenn=d+k,theposteriordistributionisproperwithprobabilityone.Then,anapplicationofLemma2ofMarchevandHobert[ 28 ]showsthatfor-almostallytheposteriorisproperfornd+k.Wewillnowshowthattheposteriordistributionisimproperwhenn
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2:uii>0;8i=1;:::;dg:NoticethatthecolumnsofLformabasisofRd.Sothereexistsconstantsb1;b2;:::;bdwithbi6=0forsomeisuchthatx0=Pdi=1bili.Supposei0=minfi2f1;2;:::;dg:bi6=0g.NowconsiderthetransformationL=(l1;l2;:::;ld)!O=(o1;o2;:::;od)whereoi=li8i6=i0andoi0=x0i.e.,O=LA,whereA=0BBBBBBBBBBBBBB@10000100...00bi00...00bd11CCCCCCCCCCCCCCA:ThentheJacobianofthetransformationisjjAjdj=jbi0jd[ 29 ,p.36].Notethatli0=1 2dXi=1lTilidYi=1lnkiiidL=jbi0jd 2Xi6=i0oTioi+1 2Xii01+b2i 2:vii>0;8i6=i0andbi0vi0i0>0g:ByFubini'stheoremwecanrearrangetheorderofintegration.Noticethatoi0doesnotappearintheexponentialtermandtheonlyterminvolvingoi0i0intheaboveintegralis 56

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AswementionedintheintroductionthatFernandezandSteel[ 12 ]gaveaproofofproprietyoftheposteriordensity(;jy).Abyproductofouralternativeproofisamethodofexactsamplingfrom(;jy)intheparticularcasewhennisd+k.Wedescribethemethodnow.Let(q;;jy)bethejointposteriordensityof(q;;)givenby(q;;jy)=f(y;qj;)(;) 4{5 )itiseasytoseethat 4{10 )weknowthat(qjy)=Qni=1h(qi).So,inthiscaseanexactdrawfrom(;jy)canbemadeusingthefollowingthreesteps: (i) Drawq1;q2;:::qnindependentlywhereqih(qi). (ii) DrawIWd[nk;(yTQ1yyTQ1X(XTQ1X)1XTQ1y)1] (iii) DrawTNd;k(yTQ1X(XTQ1X)1;;(XTQ1X)1) 57

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36 ])havefunctionsforgeneratingrandommatricesfromtheInverseWishartdistribution.OnewaytogenerateZNm;r(;;)istorstindependentlydrawZiindNr(i;)whereTiistheithrowoffori=1;:::;m.ThentakeZ=1 2266666664ZT1ZT2...ZTm377777775: 4{3 ).WerstdeveloptheDAalgorithminSection 4.3.1 usingthelatentdataq=(q1;q2;;qn)andthejointposteriordensityof(q;;jy).WethenderivetheHaarPX-DAalgorithminsection 4.3.2 .Inthespecialcase,whentheobservations,yi's,areassumedtobefromamultivariateStudent'stlocation-scalemodel,vanDykandMeng[ 51 ]developedthemarginalaugmentationalgorithm,whichisamodiedversionofthestandarddataaugmentationalgorithm,forthedensity( 4{3 ).HobertandMarchev[ 17 ]haveshownthatwhenthegroupstructureexploitedbyLiuandWu[ 27 ]exists,marginalaugmentationalgorithm(withleft-Haarmeasurefortheworkingprior)isexactlysameasthe LiuandWu 's[ 1999 ]HaarPX-DAalgorithm.Insection 4.3.2 weshowthatasimilargroupstructurecanbeestablishedforanalyzingtheposteriordensityin( 4{3 )andsomarginalaugmentationalgorithmisthesameastheHaarPX-DAalgorithminourcase.

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4.2 ,weknowthatconditionallyj;q;yfollowsaMatrixNormaldistributionandtheconditionaldistributionofjq;yisanInverseWishartdistribution.Conditionalon(;;y)qi'sareindependentwithqij;;yindh(qi)qd 2:

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HobertandMarchev 's[ 2008 ]Section4.3.FromSection 4.2 ,weknowthat(qjy)/jXTQ1Xjd q:d q:nk g=c3Z10nYi=11 :Inordertoshowthatm(q)<1,supposex1 ;x>0andconsiderthestandardnoninformativeprior,1=,forthescaleparameter,.ThenthecorrespondingposteriordistributionispropersinceZ101 d =1 )=1

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28 ]itfollowsthatR10Qni=11 <1forallx1;x2;:::;xn>0andn1sinceQni=11 .Hence,itfollowsthatm(q)<1.ConsiderthefollowingunivariatedensityonR+eq(g)=gn1 (i) Drawq(qj0;0;y) (ii) Drawgeq(g)andsetq0=gq Draw;(;jq0;y)TheMarkovtransitiondensityofthePX-DAalgorithmcanbewrittenas~k(;j0;0)=ZRn+ZRn+(;jq0;y)R(q;dq0)(qj0;0;y)dq;whereR(q;dq0)istheMarkovtransitiondensityinducedbythestep2,thattakesq!q0=gq.InthespecialcasewhenwehavemultivariateStudent'stdata,itiseasytoseethatthedensityeq(g)isGamma(n 61

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jPj:SincePisap.d.matrix,jPj>0.Similarly,sincePxxTisap.s.d.matrix,jPxxTj0.ThenfromtheaboveidentityitfollowsthatxTP1x1. ThefollowinglemmaestablishesthedriftinequalityfortheDAalgorithm. +d2V(0;0)+n+dk +d2:

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Tocalculatetheaboveconditionalexpectationsweneedthecorrespondingconditionaldistributions.Therequiredconditionaldistributions,asderivedintheprevioussections,arethefollowingTj;q;yNd;k(T;;)jq;yIWdnk;yTQ1yT11andqij;;yind+d 2i=1;2;:::;n:Startingwiththeinnermostexpectation,wehaveE(V(;)j;q;y)=EnXi=1yTi1yi2nXi=1yTi1Txi+nXi=1xTi1Txi;q;y=nXi=1yTi1yi2nXi=1EyTi1Txi;q;y+nXi=1ExTi1Txi;q;y:Tocalculatetheaboveexpectationsweusethefollowingpropertyofmatrixnormaldistribution.LetZNm;r(;;).IfCandDaretwomatricesofdimensionnmandrsrespectivelythenCZDNn;s(CD;CC0;D0D)[ 2 ,chap17].So,1 2T;q;yNd;k(1 2T;I;).Notethat,1T=1 2TT(1 2T.Therefore,1T;q;yWk(d;;1T),wherebyVWp(m;;)wemeanthatVhaspdimensionalnoncentralWishartdistributionwithmdegreesoffreedom,covariancematrixandwithnoncentralityparameter[ 2 ,chap17].Inthiscase,E(V)=m+.If=0,wesaythatVhasacentralWishartdistributionandwewriteVWp(m;).So,wehave 63

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4{12 ),weusethefactthatifXIWp(m;),thenX1Wp(m;)andE(X1)=m[ 29 ,p.85].Thus,E[E(V(;)j;q;y)jq;y]=nXi=1(yiTxi)TE1jq;y(yiTxi)+dnXi=1xTixi=(nk)nXi=1(yiTxi)T(yTQ1yT1)1(yiTxi)+dnXi=1xTixi:

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1 yields (yiTxi)T(yTQ1yT1)1(yiTxi)1 1 gives Therefore,wegetE(V(;)jq;y)(n+dk)nXi=11 2i=1;2;:::;n:So,usingthefactthatifwGamma(a;b)thenE(1 a1,nally,wehaveE(V(;)j0;0)n+dk +d2nXi=1(yi0Txi)T(0)1(yi0Txi)+;whichprovesthelemma. Thefollowinglemmaestablishesanassociatedminorizationcondition. 65

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llog1+l and(a;b;x)denotestheGamma(a,b)densityevaluatedatthepointx. Proof. 2nYi=1inf(0;0)2S+d 2nYi=1inf(0;0)2Si+d 2:Thenby Hobert 's[ 2001 ]Lemma1,itstraightforwardlyfollowsthat(qj0;0;y)nYi=1g(qi)8(0;0)2S:

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+d2isstrictlylessthan1.Westatethisinthefollowingtheorem. +d2<1i.e.,n<+k2. 's[ 2008 ]Proposition6showsthattheHaarPX-DAalgorithmisatleastasecient(ineciencyordering)astheDAalgorithm.UsingsimilarargumentsasinCorollary1,wecanshowthatgeometricergodicityoftheDAalgorithmimpliesthatoftheHaarPX-DAalgorithm.Hencewehavethefollowingcorollary. 2 matcheswith MarchevandHobert 's[ 2004 ]Theorem10inthecasewhenk=1.WealsocanprovethegeometricergodicityoftheHaarPX-DAalgorithmbydirectlyestablishingadriftandminorizationconditionforit.WeactuallycanusethesamedriftfunctionV(;)=Pni=1(yiTxi)T1(yiTxi)toestablishadriftconditionfortheHaarPX-DAalgorithm. (n2)(+d2)V(0;0)+n(n+dk) (n2)(+d2):

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4.3.2 weknowthatq0=gqwheregjqGamma(n 4{13 )andthenstraightforwardalgebrashowsthatE(V(;)jq0;y)=(nk) gnXi=1xTixi:So,E(V(;)jq;y)=(nk)nXi=1(yiTxi)T(yTQ1yT1)1(yiTxi)+dnXi=1xTixiE1 4{14 )and( 4{15 )itthenfollowsthatE(V(;)jq;y)(n+dk) 2i=1;2;:::;n;wehaveE(V(;)j0;0)(n+dk) +d2nXi=1Xj6=i+(0Txiyi)T(0)1(0Txiyi)

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OurminorizationconditionfortheDAalgorithmstraightforwardlygeneralizestoaminorizationconditionforthePX-DAalgorithm. 3 .Together,Lemmas 4 and 5 provethefollowingtheorem. (n2)(+d2)<1.AsacorollaryofTheorem 4 (seeCorollary 2 )weknowthatthePX-DAalgorithmisgeometricallyergodicifn+dk +d2<1.AtrstitmightappearthatTheorem 5 isabetterresultthanCorollary 2 .But,wenowshowthatitcanneverhappenthatbothofthefollowinginequalitiesholdtogether +d21;(n+dk)(+d)(n1) (n2)(+d2)<1:(4{16)Notethat(n+dk)(+d)(n1) (n2)(+d2)
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4{17 )holdsifandonlyifn+k2andn<1+2 4 andTheorem 5 areupshotsoftheparticulardriftfunctionV(;)andtheinequalities( 4{14 )and( 4{15 )thatweusedtoprovethedriftconditions.Inouropinion,tomakesubstantialimprovementofTheorem 4 andTheorem 5 ,weeitherhavetoconsideradierentdriftfunctionorresorttoaaltogetherdierenttechniqueofprovinggeometricergodicityotherthanestablishingdriftandminorizationcondition.Eitherofthesetwowouldrequireustostartfromscratch. 70

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(h;g) 2.Iftheresultingnormedspaceiscomplete,itiscalledaHilbertspace.LetHbeaHilbertspaceoverCandT:H!Hbealineartransformation,calledalinearoperator.TheoperatorTissaidtobeboundedifthereexistsanM>0suchthatkThkMkhkforallh2H,wherethenorm,kk,isasdenedabove.LetB(H)bethecollectionofalllinear,boundedoperatorsfromHintoH.ForT2B(H),denetheoperatornormofTbykTk=supkThk khk:h2H;h6=0:ItiseasytoseethatkTk=supfkThk:h2H;khk1g=supfkThk:h2H;khk=1g:

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47 ,p.311].Somepropertiesofadjointoperatorsarelistedbelow:kTk=kTkT=TkTTk=kTk2andifSalsobelongstoB(H)then(ST)=TS:Wenowcandenedierenttypesofoperators.AnoperatorT2B(H)issaidtobe 4 weknowthat(h;Th)= (Th;h). 72

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47 ,p.310]. Proof. SupposeT2B(H).TheoperatorTisinvertibleifthereisanS2B(H)suchthatTS=I=ST.TheoperatorSiscalledtheinverseofTandwewriteS=T1.SupposeR(T)denotestherangeofT,i.e.,R(T)=fTh:h2Hg.ThenTisinvertibleifandonlyifR(T)=HandTisone-to-one.Thespectrum,(T),oftheoperatorT2B(H)isdenedasfollows(T)=f:TIisnotinvertibleg:Thus2(T)ifandonlyifatleastoneofthefollowingtwostatementsistrue: (i) TherangeofTIisnotallofHi.e.,TIisnotonto. (ii) TheoperatorTIisnotone-to-one. 73

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9 ,p.83]. n: Proof.

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4 tellsusthat(T)isaboundedsetforanyboundedlinearoperatorTbecause(T)f2C:jjkTkg: Proof. ItcanalsobeprovedthatforT2B(H);H6=f0g,thespectrumofT,(T),isnotempty[ 47 ,p.253].Thespectralradius,rT,ofTisdenedasrT=supfjj:2(T)g:

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13 isofcourseaboundedoperatoranditcanbeprovedthatkf(T)ksupfjf()j:2(T)g.Wealsohavethefollowingtheoremwhichdeterminesthespectrumoftheoperatorf(T). 37 ]givesanice,concisedescriptionofcompactoperators.ForacompactoperatorT,thespectrum,(T),isatmostcountable,andhasatmostonelimitpoint,namely,0.Also,anynon-zerovalueinthespectrumisnecessarilyaneigenvalueofT.Westatetheseresultsinthefollowingtheorem([ 8 ]p.214,[ 9 ]chapter5). 79

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47 ]p.112,[ 8 ]p.267).Let(;B();)beameasurespace.Suppose,t:!CbesuchthatZZjt(x;y)j2d(x)d(y)<1:ThentheassociatedHilbert-SchmidtintegraloperatoristheoperatorT:L2(;C)!L2(;C)givenby(Tf)(x)=Zt(x;y)f(y)d(y):

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MeynandTweedie 's[ 1993 ]Proposition17.4.1itfollowsthatforHarrisergodicMarkovchainstheonlyfunctionsinL2()satisfyingtheequationlPf=0are-almosteverywhereconstant.SinceL20()doesnotincludefunctionswhichare-almosteverywhereconstant(non-zero)andsinceweconsiderHarrisergodicMarkovchains,theoperatorlPisone-to-oneinourcase.But,lPmightnotbeanontooperatorandsoitmightnotbeinvertible.FromthedenitionofspectrumofanoperatorweknowthatlPisinvertiblei162(P).Since(P)isaclosedsetitfollowsthat162(P)ifandonlyifsup(P)<1.HencetheLaplacianoperator,lP,isinvertibleisup(P)<1.RecallfromChapter2thatwesayaCLTholdsfor 13 ]thataCLTholdsforanyfunctionflyinginR(lP),therangeoflP,andthecorrespondingasymptoticvarianceintheCLTisgivenbyv(f;P)=kgk2kPgk2;wheref=lPgg2L20().So,iflPisinvertible,thenCLTholdsforeveryfunctionf2L20().Also,fromTheorem 7 weknowthatifkPk<1,thenlPisalwaysinvertible.Hence,wehavethefollowingproposition.

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32 ].From[ 13 ]itthenfollowsthatCLTholdsforallf2D(l1P)andbylemma3.2ofMiraandGeyer[ 32 ]wehavev(f;P)=(f;[2l1PI]f);8f2D(l1P):InChapter2,wedenedtheeciencyorderingdueto[ 32 ].TheproblemwitheciencyorderingisthatevenwhenweconcludePEQ,itmighthappenthatv(f;P)=v(f;Q)foreveryf2L2()thatareofinterest.Inthischapter,wediscussconditionsthatguarantee 5{1 )implies( 5{2 ).But,( 5{2 )doesnotnecessarilyimply( 5{1 ).LetlPandlQbetheLaplacianoperatorscorrespondingtotheMarkovoperatorsPandQ.IfweassumethatbothlPandlQareinvertiblethenitfollowsthatD(l1P)=L20()=D(l1Q).Hence,wehavev(f;P)=(f;[2l1PI]f);8f2L20()andv(f;Q)=(f;[2l1QI]f);8f2L20(): 2I0 83

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2I0: 32 ]. 3 ,ifwereplacetheinequalitiesbystrictinequalitiesandcby1,wegetconditionsforstricteciency.FortherestofthissectionweassumethatwehavereversibleMarkovchains.WeknowthattheMarkovtransitionoperator,P,ofareversibleMarkovchainisself-adjoint.So,wecanusespectraltheorytoanalyzereversibleMarkovchains.Let 24 ]showthatifv(f;P)in( 5{3 )isnite,thenthereisaCLTforfwiththeasymptoticvariance,v(f;P)givenby( 5{3 ).Nowonwards,wedenoteEf;f()byEfP().RecallfromChapter2thatPissaidtobebetterthanQintheeciencyorderingwrittenPEQ,ifv(f;P)v(f;Q)foreveryf2L2().ItcanbeeasilyshownthatthedenitionofeciencyorderingisunchangedifthespaceL2()isreplacedwithL20().Itistemptingtoconcludefrom( 5{3 )andthespectraltheoremthatPEQiI+Q IQI+P IP.But,since(P)[1;1],thefunctionh(x)=1+x 84

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32 ]. 35 ],Tierney[ 50 ]).Also,Tierney[ 50 ]showsthatPeskunorderingimpliescovarianceordering.So,P1I,i.e.,lP=IPisapositiveoperator.Fromourdiscussionintheprevioussectionitthenfollowsthatthereexistsauniquesquarerootl1 2PoflP.KipnisandVaradhan[ 24 ]showthatv(f;P)<1ifandonlyiff2R(l1 2P).Notethat,thisdoesnotcontradicttheresultofGordinandLifsic[ 13 ]sinceR(lP)R(l1 2P).MiraandGeyer[ 32 ]provethatP1QifandonlyifPEQ.WegiveanalternativeproofforP1QimpliesPEQ.ThefollowingtheoremisfromBendatandSherman[ 3 ][seealso 32 ,Theorem4.1]. 3 ]mentionthatthefunctionh(x)=ax+b cx+dwithadbc>0satisestheconditionofTheorem 16 eitherinx>d corx
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cx+d,wehaveadbc=2(>0)andd c=1 16 .Since,weareassumingthatP1Qi.e.,QP,byTheorem 16 wehaveh(Q)h(P)for2(0;1):For2(0;1),h(x)isalsoaboundedBorelfunctionon[-1,1].Sowecanusethespectraltheoremtodenetheoperatorh(P).Byspectraltheorem,wehave(h(P)f;f)=Z(P)h()EfP(d):Let+(P)=(P)T[0;1]and(P)=(P)T[1;0)thenwecanwriteZ(P)h()EfP(d)=Z+(P)h()EfP(d)+Z(P)h()EfP(d):Notethat,d dh()=2

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Proof. 16 wehaveQP)I+Q IQI+P IP:Since(P)<1,thefunctionh(x)iscontinuouson(P).ByTheorem 14 ,weknowthath(P)isaboundedoperator.SinceweconsiderthescalareldtobeR,byTheorem 14 ,wealsoknowthath(P)isself-adjoint.Letg(x)=x1 16 ongtoshowI+Q IQI+P IP)QP:Hence,wegetQP,I+Q IQI+P IP:Thefunctionh(x)=1+x IP.Then,by( 5{3 )wehaveI+Q IQI+P IP,PEQ:Hence,P1Q,QP,I+Q IQI+P IP,PEQ:

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2 weknowthatifkPk<1,thenCLTholdsforeverysquareintegrablefunctionf.ItiseasytoseethatifPisacompactoperatorthenkPk<1.So,inordertoestablishCLTforaMarkovchainwecanshowthatthecorrespondingMarkovtransitionoperator,P,iscompact.InSection 5.1 ,wementionedthatanyHilbertSchmidtoperatoriscompact.RecallfromChapter2thatifareversibleMarkovchainisgeometricallyergodic,thentheCLTholdsforeverysquareintegrablefunctionf[ 41 ].AlsowementionedbeforethatareversibleMarkovchain,P,isgeometricallyergodicifandonlyifkPk<1[ 41 44 ].SchervishandCarlin[ 48 ]andLiuetal.[ 26 ])haveprovedgeometricergodicityofcertainMarkovchainsbyestablishingthatthecorrespondingMarkovoperator,P,isHilbertSchmidt. 88

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Herewestate ChenandShao 's[ 2000 ]necessaryandsucientconditionsforc1(y)<1aswellasasimplemethodforcheckingtheseconditions.LetXdenotethenpmatrixwhoseithrowisxTiandletWdenoteannpmatrixwhoseithrowiswTi,wherewi=8><>:xiifyi=0xiifyi=1: 6 ]Thefunctionc1(y)isniteifandonlyif (i) thedesignmatrixXhasfullcolumnrank,and (ii) thereexistsavectora=(a1;:::;an)TwithstrictlypositivecomponentssuchthatWTa=0.AssumingthatXhasfullcolumnrank,thesecondconditionofProposition 5 canbestraightforwardlycheckedwithasimplelinearprogramimplementableintheRprogramminglanguage[ 18 ]usingthe\simplex"functionfromthe\boot"library.Let1andJdenoteacolumnvectorandamatrixof1s,respectively.Thelinearprogramcallsformaximizing1Tasubjectto 5 issatisedandc1(y)<1.Moreover,itisstraightforwardtoshowthatifacontainsoneormorezeros,thentheredoesnotexistanawithallpositiveelementssuchthatWTa=0,soc1(y)=1. 89

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